arXiv:math/0303046v2 [math.AT] 14 Aug 2003 Noncommutative localization in topology Andrew Ranicki Introduction The topological applications of the Cohn noncommutative localization con- sidered in this paper deal with spaces (especially manifolds) with infinite fundamental group, and involve localizations of infinite group rings and related triangular matrix rings. Algebraists have usually considered non- commutative localization of rather better behaved rings, so the topological applications require new algebraic techniques. Part 1 is a brief survey of the applications of noncommutative localization to topology: finitely dominated spaces, codimension 1 and 2 embeddings (knots and links), homology surgery theory, open book decompositions and circle-valued Morse theory. These applications involve chain complexes and the algebraic K- and L-theory of the noncommutative localization of group rings. Part 2 is a report on work on chain complexes over generalized free prod- ucts and the related algebraic K- and L-theory, from the point of view of noncommutative localization of triangular matrix rings. Following Bergman and Schofield, a generalized free product of rings can be constructed as a noncommutative localization of a triangular matrix ring. The novelty here is the explicit connection to the algebraic topology of manifolds with a gen- eralized free product structure realized by a codimension 1 submanifold, leading to noncommutative localization proofs of the results of Waldhausen and Cappell on the algebraic K- and L-theory of generalized free prod- ucts. In a sense, this is more in the nature of an application of topology to noncommutative localization! But this algebra has in turn topological applications, since in dimensions 5 the surgery classification of manifolds within a homotopy type reduces to algebra. 1
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Aug
200
3
Noncommutative localization in topology
Andrew Ranicki
Introduction
The topological applications of the Cohn noncommutative localization con-sidered in this paper deal with spaces (especially manifolds) with infinitefundamental group and involve localizations of infinite group rings andrelated triangular matrix rings Algebraists have usually considered non-commutative localization of rather better behaved rings so the topologicalapplications require new algebraic techniques
Part 1 is a brief survey of the applications of noncommutative localizationto topology finitely dominated spaces codimension 1 and 2 embeddings(knots and links) homology surgery theory open book decompositions andcircle-valued Morse theory These applications involve chain complexes andthe algebraic K- and L-theory of the noncommutative localization of grouprings
Part 2 is a report on work on chain complexes over generalized free prod-ucts and the related algebraic K- and L-theory from the point of view ofnoncommutative localization of triangular matrix rings Following Bergmanand Schofield a generalized free product of rings can be constructed as anoncommutative localization of a triangular matrix ring The novelty hereis the explicit connection to the algebraic topology of manifolds with a gen-eralized free product structure realized by a codimension 1 submanifoldleading to noncommutative localization proofs of the results of Waldhausenand Cappell on the algebraic K- and L-theory of generalized free prod-ucts In a sense this is more in the nature of an application of topologyto noncommutative localization But this algebra has in turn topologicalapplications since in dimensions gt 5 the surgery classification of manifoldswithin a homotopy type reduces to algebra
1
Part 1 A survey of applications
We start by recalling the universal noncommutative localization of PMCohn[5] Let A be a ring and let Σ = s P rarr Q be a set of morphismof fg projective A-modules A ring morphism A rarr R is Σ-invertingif for every s isin Σ the induced morphism of fg projective R-modules1otimes s RotimesAP rarr RotimesAQ is an isomorphism The noncommutative localiza-tion A rarr Σminus1A is Σ-inverting and has the universal property that any Σ-inverting ring morphism A rarr R has a unique factorization A rarr Σminus1A rarr RThe applications to topology involve homology with coefficients in a non-commutative localization Σminus1A
Homology with coefficients is defined as follows Let X be a connectedtopological space with universal cover X and let the fundamental groupπ1(X) act on the left of X so that the (singular) chain complex S(X) is a freeleft Z[π1(X)]-module complex Given a morphism of rings F Z[π1(X)] rarr Λdefine the Λ-coefficient homology of X to be
Hlowast(X Λ) = Hlowast(ΛotimesZ[π1(X)] S(X))
If X is a CW complex then S(X) is chain equivalent to the cellular freeZ[π1(X)]-module chain complex C(X) with one generator in degree r foreach r-cell of X and
Hlowast(X Λ) = Hlowast(ΛotimesZ[π1(X)] C(X))
11 Finite domination
A topological spaceX is finitely dominated if there exist a finite CW complexK maps f X rarr K g K rarr X and a homotopy gf ≃ 1 X rarr XThe finiteness obstruction of Wall [31] is a reduced projective class [X] isinK0(Z[π1(X)]) such that [X] = 0 if and only if X is homotopy equivalent toa finite CW complex
In the applications of the finiteness obstruction to manifold topologyX = M is an infinite cyclic cover of a compact manifold M ndash see Chapter 17of Hughes and Ranicki [13] for the geometric wrapping up procedure whichshows that in dimension gt 5 every tame manifold end has a neighbourhoodwhich is a finitely dominated infinite cyclic cover M of a compact manifoldM Let f M rarr S1 be a classifying map so that M = flowastR and letM
+= flowastR+ The finiteness obstruction [M
+] isin K0(Z[π1(M )]) is the end
2
obstruction of Siebenmann [27] such that [M+] = 0 if and only if the tame
end can be closed ie compactified by a manifold with boundary
Given a ring A let Ω be the set of square matrices ω isin Mr(A[z zminus1])
over the Laurent polynomial extension A[z zminus1] such that the A-module
P = coker(ω A[z zminus1]r rarr A[z zminus1]r)
is fg projective The noncommutative Fredholm localization Ωminus1A[z zminus1]has the universal property that a finite fg free A[z zminus1]-module chaincomplex C is A-module chain equivalent to a finite fg projective A-modulechain complex if and only if Hlowast(Ω
Let M be a connected finite CW complex with a connected infinite cycliccover M The fundamental group π1(M) fits into an extension
1 rarr π1(M ) rarr π1(M) rarr Z rarr 1
and Z[π1(M)] is a twisted Laurent polynomial extension
Z[π1(M)] = Z[π1(M )]α[z zminus1]
withα π1(M ) rarr π1(M) g 7rarr zminus1gz
the monodromy automorphism For the sake of simplicity only the untwistedcase α = 1 will be considered here so that π1(M) = π1(M )timesZ The infinitecyclic cover M is finitely dominated if and only if Hlowast(M Ωminus1Z[π1(M)]) = 0with A = Z[π1(M)] and Z[π1(M)] = A[z zminus1] The Farrell-Siebenmann ob-struction Φ(M) isin Wh(π1(M)) of an n-dimensional manifold M with finitelydominated infinite cyclic cover M is such that Φ(M) = 0 if (and for n gt 6only if) M is a fibre bundle over S1 ndash see [21 Proposition 1516] for theexpression of Φ(M) in terms of the Ωminus1Z[π1(M)]-coefficient Reidemeister-Whitehead torsion
Surgery theory asks whether a homotopy equivalence of manifolds is homo-topic (or h-cobordant) to a homeomorphism ndash in general the answer is noThere are obstructions in the topological K-theory of vector bundles in the
3
algebraic K-theory of modules and in the algebraic L-theory of quadraticforms The algebraic K-theory obstruction lives in the Whitehead groupWh(π) of the fundamental group π The L-theory obstruction lives in oneof the surgery groups Llowast(Z[π]) of Wall [32] and is defined when the topo-logical and algebraic K-theory obstructions vanish The groups Llowast(Λ) aredefined for any ring with involution Λ to be the generalized Witt groups ofnonsingular quadratic forms over Λ For manifolds of dimension gt 5 thevanishing of the algebraic obstructions is both a necessary and sufficientcondition for deforming a homotopy equivalence to a homeomorphism SeeRanicki [20] for the reduction of the Browder-Novikov-Sullivan-Wall surgerytheory to algebra
A homotopy equivalence of m-dimensional manifolds f M prime rarr M splits
along a submanifold Nn sub Mm if f is homotopic to a map (also denoted byf) such that N prime = fminus1(N) sub M prime is also a submanifold and the restrictionf | N prime rarr N is also a homotopy equivalence For codimension m minus n gt 3the splitting obstruction is just the ordinary surgery obstruction σlowast(f |) isinLm(Z[π1(N)]) For codimension m minus n = 1 2 the splitting obstructionsinvolve the interplay of the knotting properties of codimension (m minus n)submanifolds and Mayer-Vietoris-type decompositions of the algebraic K-and L-groups of Z[π1(M)] in terms of the groups of Z[π1(N)] Z[π1(MN)]
In the case m minus n = 1 π1(M) is a generalized free product ie eitheran amalgamated free product or an HNN extension by the Seifert-vanKampen theorem Codimension 1 splitting theorems and the algebraic K-and L-theory of generalized free products are a major ingredient of high-dimensional manifold topology featuring in the work of Stallings Brow-der Novikov Wall Siebenmann Farrell Hsiang Shaneson Casson Wald-hausen Cappell and the author Noncommutative localization providesa systematic development of this algebra using the intuition afforded by thetopological applications ndash see Part 2 below for a more detailed discussion
13 Homology surgery theory
For a morphism of rings with involution F Z[π] rarr Λ Cappell and Shaneson[3] considered the problem of whether a Λ-coefficient homology equivalenceof manifolds with fundamental group π is H-cobordant to a homeomor-phism Again the answer is no in general with obstructions in the topolog-ical K-theory of vector bundles and in the homology surgery groups Γlowast(F)which are generalized Witt groups of Λ-nonsingular quadratic forms over
4
Z[π] Vogel [28] [29] identified the Λ-coefficient homology surgery groupswith the ordinary L-groups of the localization Σminus1Z[π] of Z[π] inverting theset Σ of Λ-invertible square matrices over Z[π]
Γlowast(F) = Llowast(Σminus1Z[π])
and identified the relative L-groups Llowast(Z[π] rarr Σminus1Z[π]) in the localizationexact sequence
with generalized Witt groups Llowast(Z[π]Σ) of nonsingular Σminus1Z[π]Z[π])-valued quadratic linking forms on Σ-torsion Z[π]-modules of homologicaldimension 1
14 Codimension 2 embeddings
Suppose given a codimension 2 embedding Nn sub Mn+2 such as a knot orlink Let Σminus1A be the localization of A = Z[π1(MN)] inverting the set Σof matrices over A which become invertible over Z[π1(M)] By Alexanderduality the Σminus1A-coefficient homology modules
are determined by the homotopy class of the inclusion N sub M The A-coefficient homology groups Hlowast(MN A) and their Poincare duality prop-erties reflect more subtle invariants of N sub M such as knotting See Ranicki[21] for a general account of high-dimensional codimension 2 embedding the-ory including some of the applications of noncommutative localization
15 Open books
An (n + 2)-dimensional manifold Mn+2 is an open book if there exists acodimension 2 submanifold Nn sub Mn+2 such that the complement MN isa fibre bundle over S1 Every odd-dimensional manifold is an open bookQuinn [17] showed that for k gt 2 a (2k + 2)-dimensional manifold M isan open book if and only if an asymmetric form over Z[π1(M)] associatedto M represents 0 in the Witt group This obstruction was identified inRanicki [21] with an element in the L-group L2k+2(Ω
An n-dimensional micro-component boundary link is a codimension 2 embedding
Nn =⋃
micro
Sn sub Mn+2 = Sn+2
with a micro-component Seifert surface in which case the fundamental groupof the complement X = MN has a compatible surjection π1(X) rarr Fmicro
onto the free group on micro generators Duval [8] used the work of Cap-pell and Shaneson [4] and Vogel [29] to identify the cobordism group ofn-dimensional micro-component boundary links for n gt 2 with the relative L-group Ln+3(Z[Fmicro]Σ) in the localization exact sequence
with Σ the set of Z-invertible square matrices over Z[Fmicro] The even-dimensionalboundary link cobordism groups are L2lowast+1(Z[Fmicro]Σ) = 0 The cobordismclass in L2k+2(Z[Fmicro]Σ) of a (2k minus 1)-dimensional micro-component boundarylink cupmicroS
2kminus1 sub S2k+1 was identified with the Witt class of a Σminus1Z[Fmicro]Z[Fmicro]-valued nonsingular (minus1)k+1-quadratic linking form on Hk(XZ[Fmicro]) gener-alizing the Blanchfield pairing on the homology of the infinite cyclic coverof a knot The localization Σminus1Z[Fmicro] was identified by Dicks and Sontag [7]and Farber and Vogel [11] with a ring of rational functions in micro noncommut-ing variables The high odd-dimensional boundary link cobordism groupsL2lowast+2(Z[Fmicro]Σ) have been computed by Sheiham [26]
17 Circle-valued Morse theory
Novikov [15] proposed the study of the critical points of Morse functionsf M rarr S1 on compact manifolds M The lsquoNovikov complexrsquo C(Mf)over Z((z)) = Z[[z]][zminus1] has one generator for each critical point of f andthe lsquoNovikov homologyrsquo
Hlowast(C(Mf)) = Hlowast(M Z((z)))
provides lower bounds on the number of critical points of Morse functions inthe homotopy class of f generalizing the inequalities of the classical Morsetheory of real-valued functions M rarr R Suppose given a Morse functionf M rarr S1 with M = flowastR such that π1(M) = π1(M ) times Z (for the sakeof simplicity) Let Σ be the set of square matrices over Z[π1(M )][z] which
6
become invertible over Z[π1(M)] under the augmentation z 7rarr 0 There is anatural morphism from the localization to the completion
which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also
features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]
18 3- and 4-dimensional manifolds
See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds
Part 2 The algebraic K- and L-theory of general-
ized free products via noncommutative localization
A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of
the Z[π1(M)]-module chain complex C(M) of the universal cover M is the
7
assembly of an A-module chain complex constructed from the chain com-
plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24
21 The algebraic K-theory of a noncommutative localiza-
tion
Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution
0 Ps
Q T 0
such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]
was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo
TorAi (Σminus1AΣminus1A) = 0 (i gt 1)
then
(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B
(ii) the localization exact sequence extends to the higher K-groups
The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings
Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring
A =
(A1 B0 A2
)
An A-module can be written as
M =
(M1
M2
)
with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection
A1 timesA2 rarr A (a1 a2) 7rarr
(a1 00 a2
)
induces isomorphisms of algebraic K-groups
Klowast(A1)oplusKlowast(A2) sim= Klowast(A)
The columns of A are fg projective A-modules
P1 =
(A1
0
) P2 =
(BA2
)
such that
P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)
HomA(P1 P2) = B HomA(P2 P1) = 0
The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring
Σminus1A = M2(C) =
(C CC C
)
with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1
sim= Σminus1P2 The Morita equivalence
Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L
9
induces isomorphisms in algebraic K-theory
Klowast(M2(C)) sim= Klowast(C)
The composite of the functor
A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M
and the Morita equivalence is the assembly functor
A-modules rarr C-modules
M =
(M1
M2
)7rarr (C C)otimesA M
= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)
in the algebraic K-theory localization exact sequence
There are evident generalizations to k times k matrix rings for any k gt 2
23 HNN extensions
The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with
α(s)z = zβ(s) isin R lowastαβ z (s isin S)
Define the triangular 2times 2 matrix ring
A =
(R Rα oplusRβ
0 S
)
with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with
σ1 =
((1 0)0
) σ2 =
((0 1)0
) P1 =
(R0
)rarr P2 =
(Rα oplusRβ
S
)
The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies
Σminus1A = M2(R lowastαβ z)
10
Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected
M = M1 cupNtimes01 N times [0 1]
N times [0 1]
M1
By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1
Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are
injective and the universal cover M is a union
M =⋃
gisin[π1(M)π1(M1)]
gM1
11
of translates of the universal cover M1 of M1 and
g1M1 cap g2M1 =
hN if g1 cap g2z = h isin [π1(M) π1(N)]
g1M1 if g1 = g2
empty if g1 6= g2 and g1 cap g2z = empty
with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms
M zminus2M1 zminus1M1 M1 zM1 z2M1
zminus1N N zN z2N
The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps
iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)
iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)
defining a fg projective A-module chain complex
(C(M1)
C(N)
)with assembly
the cellular fg free Z[π1(M)]-module chain complex of M
coker
(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)
)
= C(M)
by the Mayer-Vietoris theorem
Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms
R rarr R lowastαβ z S rarr R lowastαβ z
A =
(R Rα oplusRβ
0 S
)rarr Σminus1A = M2(R lowastαβ z)
12
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
are just the unitary nilpotent L-groups UNillowast of Cappell [2]
References
[1] H Bass A Heller and R Swan The Whitehead group of a polynomial
extension Publ Math IHES 22 61ndash80 (1964)
[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)
[3] and J Shaneson The codimension two placement problem and
homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)
[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)
[5] P M Cohn Free rings and their relations Academic Press (1971)
[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and
Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc
18
[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)
[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)
[9] M Farber Morse-Novikov critical point theory Cohn localization and
Dirichlet units Commun Contemp Math 1 467ndash495 (1999)
[10] and A Ranicki The Morse-Novikov theory of circle-valued func-
tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)
[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)
[12] S Garoufalidis and A Kricker A rational noncommutative invariant
of boundary links httparXivmathGT0105028 (2001)
[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)
[14] A Neeman and A Ranicki Noncommutative localization and chain
complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)
[15] S P Novikov The hamiltonian formalism and a multi-valued analogue
of Morse theory Russian Math Surveys 375 1ndash56 (1982)
[16] A Pajitnov Incidence coefficients in the Novikov complex for
Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)
[17] F Quinn Open book decompositions and the bordism of automor-
phisms Topology 18 55ndash73 (1979)
[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)
[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)
19
[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)
[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)
[22] The algebraic construction of the Novikov complex of a circle-
valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)
[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111
[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)
[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-
calization J London Math Soc 64 13ndash28 (2001)
[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)
[27] L Siebenmann The obstruction to finding the boundary of an open
manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf
[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)
[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)
[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)
[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)
[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)
School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building
20
Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK
e-mail aarmathsedacuk
21
Finite domination
Codimension 1 splitting
Homology surgery theory
Codimension 2 embeddings
Open books
Boundary link cobordism
Circle-valued Morse theory
3- and 4-dimensional manifolds
The algebraic K-theory of a noncommutative localization
Matrix rings
HNN extensions
Amalgamated free products
The algebraic L-theory of a noncommutative localization
Part 1 A survey of applications
We start by recalling the universal noncommutative localization of PMCohn[5] Let A be a ring and let Σ = s P rarr Q be a set of morphismof fg projective A-modules A ring morphism A rarr R is Σ-invertingif for every s isin Σ the induced morphism of fg projective R-modules1otimes s RotimesAP rarr RotimesAQ is an isomorphism The noncommutative localiza-tion A rarr Σminus1A is Σ-inverting and has the universal property that any Σ-inverting ring morphism A rarr R has a unique factorization A rarr Σminus1A rarr RThe applications to topology involve homology with coefficients in a non-commutative localization Σminus1A
Homology with coefficients is defined as follows Let X be a connectedtopological space with universal cover X and let the fundamental groupπ1(X) act on the left of X so that the (singular) chain complex S(X) is a freeleft Z[π1(X)]-module complex Given a morphism of rings F Z[π1(X)] rarr Λdefine the Λ-coefficient homology of X to be
Hlowast(X Λ) = Hlowast(ΛotimesZ[π1(X)] S(X))
If X is a CW complex then S(X) is chain equivalent to the cellular freeZ[π1(X)]-module chain complex C(X) with one generator in degree r foreach r-cell of X and
Hlowast(X Λ) = Hlowast(ΛotimesZ[π1(X)] C(X))
11 Finite domination
A topological spaceX is finitely dominated if there exist a finite CW complexK maps f X rarr K g K rarr X and a homotopy gf ≃ 1 X rarr XThe finiteness obstruction of Wall [31] is a reduced projective class [X] isinK0(Z[π1(X)]) such that [X] = 0 if and only if X is homotopy equivalent toa finite CW complex
In the applications of the finiteness obstruction to manifold topologyX = M is an infinite cyclic cover of a compact manifold M ndash see Chapter 17of Hughes and Ranicki [13] for the geometric wrapping up procedure whichshows that in dimension gt 5 every tame manifold end has a neighbourhoodwhich is a finitely dominated infinite cyclic cover M of a compact manifoldM Let f M rarr S1 be a classifying map so that M = flowastR and letM
+= flowastR+ The finiteness obstruction [M
+] isin K0(Z[π1(M )]) is the end
2
obstruction of Siebenmann [27] such that [M+] = 0 if and only if the tame
end can be closed ie compactified by a manifold with boundary
Given a ring A let Ω be the set of square matrices ω isin Mr(A[z zminus1])
over the Laurent polynomial extension A[z zminus1] such that the A-module
P = coker(ω A[z zminus1]r rarr A[z zminus1]r)
is fg projective The noncommutative Fredholm localization Ωminus1A[z zminus1]has the universal property that a finite fg free A[z zminus1]-module chaincomplex C is A-module chain equivalent to a finite fg projective A-modulechain complex if and only if Hlowast(Ω
Let M be a connected finite CW complex with a connected infinite cycliccover M The fundamental group π1(M) fits into an extension
1 rarr π1(M ) rarr π1(M) rarr Z rarr 1
and Z[π1(M)] is a twisted Laurent polynomial extension
Z[π1(M)] = Z[π1(M )]α[z zminus1]
withα π1(M ) rarr π1(M) g 7rarr zminus1gz
the monodromy automorphism For the sake of simplicity only the untwistedcase α = 1 will be considered here so that π1(M) = π1(M )timesZ The infinitecyclic cover M is finitely dominated if and only if Hlowast(M Ωminus1Z[π1(M)]) = 0with A = Z[π1(M)] and Z[π1(M)] = A[z zminus1] The Farrell-Siebenmann ob-struction Φ(M) isin Wh(π1(M)) of an n-dimensional manifold M with finitelydominated infinite cyclic cover M is such that Φ(M) = 0 if (and for n gt 6only if) M is a fibre bundle over S1 ndash see [21 Proposition 1516] for theexpression of Φ(M) in terms of the Ωminus1Z[π1(M)]-coefficient Reidemeister-Whitehead torsion
Surgery theory asks whether a homotopy equivalence of manifolds is homo-topic (or h-cobordant) to a homeomorphism ndash in general the answer is noThere are obstructions in the topological K-theory of vector bundles in the
3
algebraic K-theory of modules and in the algebraic L-theory of quadraticforms The algebraic K-theory obstruction lives in the Whitehead groupWh(π) of the fundamental group π The L-theory obstruction lives in oneof the surgery groups Llowast(Z[π]) of Wall [32] and is defined when the topo-logical and algebraic K-theory obstructions vanish The groups Llowast(Λ) aredefined for any ring with involution Λ to be the generalized Witt groups ofnonsingular quadratic forms over Λ For manifolds of dimension gt 5 thevanishing of the algebraic obstructions is both a necessary and sufficientcondition for deforming a homotopy equivalence to a homeomorphism SeeRanicki [20] for the reduction of the Browder-Novikov-Sullivan-Wall surgerytheory to algebra
A homotopy equivalence of m-dimensional manifolds f M prime rarr M splits
along a submanifold Nn sub Mm if f is homotopic to a map (also denoted byf) such that N prime = fminus1(N) sub M prime is also a submanifold and the restrictionf | N prime rarr N is also a homotopy equivalence For codimension m minus n gt 3the splitting obstruction is just the ordinary surgery obstruction σlowast(f |) isinLm(Z[π1(N)]) For codimension m minus n = 1 2 the splitting obstructionsinvolve the interplay of the knotting properties of codimension (m minus n)submanifolds and Mayer-Vietoris-type decompositions of the algebraic K-and L-groups of Z[π1(M)] in terms of the groups of Z[π1(N)] Z[π1(MN)]
In the case m minus n = 1 π1(M) is a generalized free product ie eitheran amalgamated free product or an HNN extension by the Seifert-vanKampen theorem Codimension 1 splitting theorems and the algebraic K-and L-theory of generalized free products are a major ingredient of high-dimensional manifold topology featuring in the work of Stallings Brow-der Novikov Wall Siebenmann Farrell Hsiang Shaneson Casson Wald-hausen Cappell and the author Noncommutative localization providesa systematic development of this algebra using the intuition afforded by thetopological applications ndash see Part 2 below for a more detailed discussion
13 Homology surgery theory
For a morphism of rings with involution F Z[π] rarr Λ Cappell and Shaneson[3] considered the problem of whether a Λ-coefficient homology equivalenceof manifolds with fundamental group π is H-cobordant to a homeomor-phism Again the answer is no in general with obstructions in the topolog-ical K-theory of vector bundles and in the homology surgery groups Γlowast(F)which are generalized Witt groups of Λ-nonsingular quadratic forms over
4
Z[π] Vogel [28] [29] identified the Λ-coefficient homology surgery groupswith the ordinary L-groups of the localization Σminus1Z[π] of Z[π] inverting theset Σ of Λ-invertible square matrices over Z[π]
Γlowast(F) = Llowast(Σminus1Z[π])
and identified the relative L-groups Llowast(Z[π] rarr Σminus1Z[π]) in the localizationexact sequence
with generalized Witt groups Llowast(Z[π]Σ) of nonsingular Σminus1Z[π]Z[π])-valued quadratic linking forms on Σ-torsion Z[π]-modules of homologicaldimension 1
14 Codimension 2 embeddings
Suppose given a codimension 2 embedding Nn sub Mn+2 such as a knot orlink Let Σminus1A be the localization of A = Z[π1(MN)] inverting the set Σof matrices over A which become invertible over Z[π1(M)] By Alexanderduality the Σminus1A-coefficient homology modules
are determined by the homotopy class of the inclusion N sub M The A-coefficient homology groups Hlowast(MN A) and their Poincare duality prop-erties reflect more subtle invariants of N sub M such as knotting See Ranicki[21] for a general account of high-dimensional codimension 2 embedding the-ory including some of the applications of noncommutative localization
15 Open books
An (n + 2)-dimensional manifold Mn+2 is an open book if there exists acodimension 2 submanifold Nn sub Mn+2 such that the complement MN isa fibre bundle over S1 Every odd-dimensional manifold is an open bookQuinn [17] showed that for k gt 2 a (2k + 2)-dimensional manifold M isan open book if and only if an asymmetric form over Z[π1(M)] associatedto M represents 0 in the Witt group This obstruction was identified inRanicki [21] with an element in the L-group L2k+2(Ω
An n-dimensional micro-component boundary link is a codimension 2 embedding
Nn =⋃
micro
Sn sub Mn+2 = Sn+2
with a micro-component Seifert surface in which case the fundamental groupof the complement X = MN has a compatible surjection π1(X) rarr Fmicro
onto the free group on micro generators Duval [8] used the work of Cap-pell and Shaneson [4] and Vogel [29] to identify the cobordism group ofn-dimensional micro-component boundary links for n gt 2 with the relative L-group Ln+3(Z[Fmicro]Σ) in the localization exact sequence
with Σ the set of Z-invertible square matrices over Z[Fmicro] The even-dimensionalboundary link cobordism groups are L2lowast+1(Z[Fmicro]Σ) = 0 The cobordismclass in L2k+2(Z[Fmicro]Σ) of a (2k minus 1)-dimensional micro-component boundarylink cupmicroS
2kminus1 sub S2k+1 was identified with the Witt class of a Σminus1Z[Fmicro]Z[Fmicro]-valued nonsingular (minus1)k+1-quadratic linking form on Hk(XZ[Fmicro]) gener-alizing the Blanchfield pairing on the homology of the infinite cyclic coverof a knot The localization Σminus1Z[Fmicro] was identified by Dicks and Sontag [7]and Farber and Vogel [11] with a ring of rational functions in micro noncommut-ing variables The high odd-dimensional boundary link cobordism groupsL2lowast+2(Z[Fmicro]Σ) have been computed by Sheiham [26]
17 Circle-valued Morse theory
Novikov [15] proposed the study of the critical points of Morse functionsf M rarr S1 on compact manifolds M The lsquoNovikov complexrsquo C(Mf)over Z((z)) = Z[[z]][zminus1] has one generator for each critical point of f andthe lsquoNovikov homologyrsquo
Hlowast(C(Mf)) = Hlowast(M Z((z)))
provides lower bounds on the number of critical points of Morse functions inthe homotopy class of f generalizing the inequalities of the classical Morsetheory of real-valued functions M rarr R Suppose given a Morse functionf M rarr S1 with M = flowastR such that π1(M) = π1(M ) times Z (for the sakeof simplicity) Let Σ be the set of square matrices over Z[π1(M )][z] which
6
become invertible over Z[π1(M)] under the augmentation z 7rarr 0 There is anatural morphism from the localization to the completion
which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also
features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]
18 3- and 4-dimensional manifolds
See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds
Part 2 The algebraic K- and L-theory of general-
ized free products via noncommutative localization
A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of
the Z[π1(M)]-module chain complex C(M) of the universal cover M is the
7
assembly of an A-module chain complex constructed from the chain com-
plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24
21 The algebraic K-theory of a noncommutative localiza-
tion
Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution
0 Ps
Q T 0
such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]
was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo
TorAi (Σminus1AΣminus1A) = 0 (i gt 1)
then
(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B
(ii) the localization exact sequence extends to the higher K-groups
The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings
Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring
A =
(A1 B0 A2
)
An A-module can be written as
M =
(M1
M2
)
with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection
A1 timesA2 rarr A (a1 a2) 7rarr
(a1 00 a2
)
induces isomorphisms of algebraic K-groups
Klowast(A1)oplusKlowast(A2) sim= Klowast(A)
The columns of A are fg projective A-modules
P1 =
(A1
0
) P2 =
(BA2
)
such that
P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)
HomA(P1 P2) = B HomA(P2 P1) = 0
The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring
Σminus1A = M2(C) =
(C CC C
)
with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1
sim= Σminus1P2 The Morita equivalence
Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L
9
induces isomorphisms in algebraic K-theory
Klowast(M2(C)) sim= Klowast(C)
The composite of the functor
A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M
and the Morita equivalence is the assembly functor
A-modules rarr C-modules
M =
(M1
M2
)7rarr (C C)otimesA M
= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)
in the algebraic K-theory localization exact sequence
There are evident generalizations to k times k matrix rings for any k gt 2
23 HNN extensions
The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with
α(s)z = zβ(s) isin R lowastαβ z (s isin S)
Define the triangular 2times 2 matrix ring
A =
(R Rα oplusRβ
0 S
)
with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with
σ1 =
((1 0)0
) σ2 =
((0 1)0
) P1 =
(R0
)rarr P2 =
(Rα oplusRβ
S
)
The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies
Σminus1A = M2(R lowastαβ z)
10
Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected
M = M1 cupNtimes01 N times [0 1]
N times [0 1]
M1
By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1
Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are
injective and the universal cover M is a union
M =⋃
gisin[π1(M)π1(M1)]
gM1
11
of translates of the universal cover M1 of M1 and
g1M1 cap g2M1 =
hN if g1 cap g2z = h isin [π1(M) π1(N)]
g1M1 if g1 = g2
empty if g1 6= g2 and g1 cap g2z = empty
with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms
M zminus2M1 zminus1M1 M1 zM1 z2M1
zminus1N N zN z2N
The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps
iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)
iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)
defining a fg projective A-module chain complex
(C(M1)
C(N)
)with assembly
the cellular fg free Z[π1(M)]-module chain complex of M
coker
(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)
)
= C(M)
by the Mayer-Vietoris theorem
Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms
R rarr R lowastαβ z S rarr R lowastαβ z
A =
(R Rα oplusRβ
0 S
)rarr Σminus1A = M2(R lowastαβ z)
12
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
are just the unitary nilpotent L-groups UNillowast of Cappell [2]
References
[1] H Bass A Heller and R Swan The Whitehead group of a polynomial
extension Publ Math IHES 22 61ndash80 (1964)
[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)
[3] and J Shaneson The codimension two placement problem and
homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)
[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)
[5] P M Cohn Free rings and their relations Academic Press (1971)
[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and
Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc
18
[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)
[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)
[9] M Farber Morse-Novikov critical point theory Cohn localization and
Dirichlet units Commun Contemp Math 1 467ndash495 (1999)
[10] and A Ranicki The Morse-Novikov theory of circle-valued func-
tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)
[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)
[12] S Garoufalidis and A Kricker A rational noncommutative invariant
of boundary links httparXivmathGT0105028 (2001)
[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)
[14] A Neeman and A Ranicki Noncommutative localization and chain
complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)
[15] S P Novikov The hamiltonian formalism and a multi-valued analogue
of Morse theory Russian Math Surveys 375 1ndash56 (1982)
[16] A Pajitnov Incidence coefficients in the Novikov complex for
Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)
[17] F Quinn Open book decompositions and the bordism of automor-
phisms Topology 18 55ndash73 (1979)
[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)
[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)
19
[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)
[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)
[22] The algebraic construction of the Novikov complex of a circle-
valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)
[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111
[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)
[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-
calization J London Math Soc 64 13ndash28 (2001)
[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)
[27] L Siebenmann The obstruction to finding the boundary of an open
manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf
[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)
[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)
[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)
[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)
[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)
School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building
20
Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK
e-mail aarmathsedacuk
21
Finite domination
Codimension 1 splitting
Homology surgery theory
Codimension 2 embeddings
Open books
Boundary link cobordism
Circle-valued Morse theory
3- and 4-dimensional manifolds
The algebraic K-theory of a noncommutative localization
Matrix rings
HNN extensions
Amalgamated free products
The algebraic L-theory of a noncommutative localization
obstruction of Siebenmann [27] such that [M+] = 0 if and only if the tame
end can be closed ie compactified by a manifold with boundary
Given a ring A let Ω be the set of square matrices ω isin Mr(A[z zminus1])
over the Laurent polynomial extension A[z zminus1] such that the A-module
P = coker(ω A[z zminus1]r rarr A[z zminus1]r)
is fg projective The noncommutative Fredholm localization Ωminus1A[z zminus1]has the universal property that a finite fg free A[z zminus1]-module chaincomplex C is A-module chain equivalent to a finite fg projective A-modulechain complex if and only if Hlowast(Ω
Let M be a connected finite CW complex with a connected infinite cycliccover M The fundamental group π1(M) fits into an extension
1 rarr π1(M ) rarr π1(M) rarr Z rarr 1
and Z[π1(M)] is a twisted Laurent polynomial extension
Z[π1(M)] = Z[π1(M )]α[z zminus1]
withα π1(M ) rarr π1(M) g 7rarr zminus1gz
the monodromy automorphism For the sake of simplicity only the untwistedcase α = 1 will be considered here so that π1(M) = π1(M )timesZ The infinitecyclic cover M is finitely dominated if and only if Hlowast(M Ωminus1Z[π1(M)]) = 0with A = Z[π1(M)] and Z[π1(M)] = A[z zminus1] The Farrell-Siebenmann ob-struction Φ(M) isin Wh(π1(M)) of an n-dimensional manifold M with finitelydominated infinite cyclic cover M is such that Φ(M) = 0 if (and for n gt 6only if) M is a fibre bundle over S1 ndash see [21 Proposition 1516] for theexpression of Φ(M) in terms of the Ωminus1Z[π1(M)]-coefficient Reidemeister-Whitehead torsion
Surgery theory asks whether a homotopy equivalence of manifolds is homo-topic (or h-cobordant) to a homeomorphism ndash in general the answer is noThere are obstructions in the topological K-theory of vector bundles in the
3
algebraic K-theory of modules and in the algebraic L-theory of quadraticforms The algebraic K-theory obstruction lives in the Whitehead groupWh(π) of the fundamental group π The L-theory obstruction lives in oneof the surgery groups Llowast(Z[π]) of Wall [32] and is defined when the topo-logical and algebraic K-theory obstructions vanish The groups Llowast(Λ) aredefined for any ring with involution Λ to be the generalized Witt groups ofnonsingular quadratic forms over Λ For manifolds of dimension gt 5 thevanishing of the algebraic obstructions is both a necessary and sufficientcondition for deforming a homotopy equivalence to a homeomorphism SeeRanicki [20] for the reduction of the Browder-Novikov-Sullivan-Wall surgerytheory to algebra
A homotopy equivalence of m-dimensional manifolds f M prime rarr M splits
along a submanifold Nn sub Mm if f is homotopic to a map (also denoted byf) such that N prime = fminus1(N) sub M prime is also a submanifold and the restrictionf | N prime rarr N is also a homotopy equivalence For codimension m minus n gt 3the splitting obstruction is just the ordinary surgery obstruction σlowast(f |) isinLm(Z[π1(N)]) For codimension m minus n = 1 2 the splitting obstructionsinvolve the interplay of the knotting properties of codimension (m minus n)submanifolds and Mayer-Vietoris-type decompositions of the algebraic K-and L-groups of Z[π1(M)] in terms of the groups of Z[π1(N)] Z[π1(MN)]
In the case m minus n = 1 π1(M) is a generalized free product ie eitheran amalgamated free product or an HNN extension by the Seifert-vanKampen theorem Codimension 1 splitting theorems and the algebraic K-and L-theory of generalized free products are a major ingredient of high-dimensional manifold topology featuring in the work of Stallings Brow-der Novikov Wall Siebenmann Farrell Hsiang Shaneson Casson Wald-hausen Cappell and the author Noncommutative localization providesa systematic development of this algebra using the intuition afforded by thetopological applications ndash see Part 2 below for a more detailed discussion
13 Homology surgery theory
For a morphism of rings with involution F Z[π] rarr Λ Cappell and Shaneson[3] considered the problem of whether a Λ-coefficient homology equivalenceof manifolds with fundamental group π is H-cobordant to a homeomor-phism Again the answer is no in general with obstructions in the topolog-ical K-theory of vector bundles and in the homology surgery groups Γlowast(F)which are generalized Witt groups of Λ-nonsingular quadratic forms over
4
Z[π] Vogel [28] [29] identified the Λ-coefficient homology surgery groupswith the ordinary L-groups of the localization Σminus1Z[π] of Z[π] inverting theset Σ of Λ-invertible square matrices over Z[π]
Γlowast(F) = Llowast(Σminus1Z[π])
and identified the relative L-groups Llowast(Z[π] rarr Σminus1Z[π]) in the localizationexact sequence
with generalized Witt groups Llowast(Z[π]Σ) of nonsingular Σminus1Z[π]Z[π])-valued quadratic linking forms on Σ-torsion Z[π]-modules of homologicaldimension 1
14 Codimension 2 embeddings
Suppose given a codimension 2 embedding Nn sub Mn+2 such as a knot orlink Let Σminus1A be the localization of A = Z[π1(MN)] inverting the set Σof matrices over A which become invertible over Z[π1(M)] By Alexanderduality the Σminus1A-coefficient homology modules
are determined by the homotopy class of the inclusion N sub M The A-coefficient homology groups Hlowast(MN A) and their Poincare duality prop-erties reflect more subtle invariants of N sub M such as knotting See Ranicki[21] for a general account of high-dimensional codimension 2 embedding the-ory including some of the applications of noncommutative localization
15 Open books
An (n + 2)-dimensional manifold Mn+2 is an open book if there exists acodimension 2 submanifold Nn sub Mn+2 such that the complement MN isa fibre bundle over S1 Every odd-dimensional manifold is an open bookQuinn [17] showed that for k gt 2 a (2k + 2)-dimensional manifold M isan open book if and only if an asymmetric form over Z[π1(M)] associatedto M represents 0 in the Witt group This obstruction was identified inRanicki [21] with an element in the L-group L2k+2(Ω
An n-dimensional micro-component boundary link is a codimension 2 embedding
Nn =⋃
micro
Sn sub Mn+2 = Sn+2
with a micro-component Seifert surface in which case the fundamental groupof the complement X = MN has a compatible surjection π1(X) rarr Fmicro
onto the free group on micro generators Duval [8] used the work of Cap-pell and Shaneson [4] and Vogel [29] to identify the cobordism group ofn-dimensional micro-component boundary links for n gt 2 with the relative L-group Ln+3(Z[Fmicro]Σ) in the localization exact sequence
with Σ the set of Z-invertible square matrices over Z[Fmicro] The even-dimensionalboundary link cobordism groups are L2lowast+1(Z[Fmicro]Σ) = 0 The cobordismclass in L2k+2(Z[Fmicro]Σ) of a (2k minus 1)-dimensional micro-component boundarylink cupmicroS
2kminus1 sub S2k+1 was identified with the Witt class of a Σminus1Z[Fmicro]Z[Fmicro]-valued nonsingular (minus1)k+1-quadratic linking form on Hk(XZ[Fmicro]) gener-alizing the Blanchfield pairing on the homology of the infinite cyclic coverof a knot The localization Σminus1Z[Fmicro] was identified by Dicks and Sontag [7]and Farber and Vogel [11] with a ring of rational functions in micro noncommut-ing variables The high odd-dimensional boundary link cobordism groupsL2lowast+2(Z[Fmicro]Σ) have been computed by Sheiham [26]
17 Circle-valued Morse theory
Novikov [15] proposed the study of the critical points of Morse functionsf M rarr S1 on compact manifolds M The lsquoNovikov complexrsquo C(Mf)over Z((z)) = Z[[z]][zminus1] has one generator for each critical point of f andthe lsquoNovikov homologyrsquo
Hlowast(C(Mf)) = Hlowast(M Z((z)))
provides lower bounds on the number of critical points of Morse functions inthe homotopy class of f generalizing the inequalities of the classical Morsetheory of real-valued functions M rarr R Suppose given a Morse functionf M rarr S1 with M = flowastR such that π1(M) = π1(M ) times Z (for the sakeof simplicity) Let Σ be the set of square matrices over Z[π1(M )][z] which
6
become invertible over Z[π1(M)] under the augmentation z 7rarr 0 There is anatural morphism from the localization to the completion
which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also
features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]
18 3- and 4-dimensional manifolds
See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds
Part 2 The algebraic K- and L-theory of general-
ized free products via noncommutative localization
A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of
the Z[π1(M)]-module chain complex C(M) of the universal cover M is the
7
assembly of an A-module chain complex constructed from the chain com-
plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24
21 The algebraic K-theory of a noncommutative localiza-
tion
Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution
0 Ps
Q T 0
such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]
was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo
TorAi (Σminus1AΣminus1A) = 0 (i gt 1)
then
(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B
(ii) the localization exact sequence extends to the higher K-groups
The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings
Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring
A =
(A1 B0 A2
)
An A-module can be written as
M =
(M1
M2
)
with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection
A1 timesA2 rarr A (a1 a2) 7rarr
(a1 00 a2
)
induces isomorphisms of algebraic K-groups
Klowast(A1)oplusKlowast(A2) sim= Klowast(A)
The columns of A are fg projective A-modules
P1 =
(A1
0
) P2 =
(BA2
)
such that
P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)
HomA(P1 P2) = B HomA(P2 P1) = 0
The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring
Σminus1A = M2(C) =
(C CC C
)
with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1
sim= Σminus1P2 The Morita equivalence
Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L
9
induces isomorphisms in algebraic K-theory
Klowast(M2(C)) sim= Klowast(C)
The composite of the functor
A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M
and the Morita equivalence is the assembly functor
A-modules rarr C-modules
M =
(M1
M2
)7rarr (C C)otimesA M
= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)
in the algebraic K-theory localization exact sequence
There are evident generalizations to k times k matrix rings for any k gt 2
23 HNN extensions
The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with
α(s)z = zβ(s) isin R lowastαβ z (s isin S)
Define the triangular 2times 2 matrix ring
A =
(R Rα oplusRβ
0 S
)
with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with
σ1 =
((1 0)0
) σ2 =
((0 1)0
) P1 =
(R0
)rarr P2 =
(Rα oplusRβ
S
)
The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies
Σminus1A = M2(R lowastαβ z)
10
Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected
M = M1 cupNtimes01 N times [0 1]
N times [0 1]
M1
By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1
Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are
injective and the universal cover M is a union
M =⋃
gisin[π1(M)π1(M1)]
gM1
11
of translates of the universal cover M1 of M1 and
g1M1 cap g2M1 =
hN if g1 cap g2z = h isin [π1(M) π1(N)]
g1M1 if g1 = g2
empty if g1 6= g2 and g1 cap g2z = empty
with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms
M zminus2M1 zminus1M1 M1 zM1 z2M1
zminus1N N zN z2N
The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps
iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)
iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)
defining a fg projective A-module chain complex
(C(M1)
C(N)
)with assembly
the cellular fg free Z[π1(M)]-module chain complex of M
coker
(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)
)
= C(M)
by the Mayer-Vietoris theorem
Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms
R rarr R lowastαβ z S rarr R lowastαβ z
A =
(R Rα oplusRβ
0 S
)rarr Σminus1A = M2(R lowastαβ z)
12
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
are just the unitary nilpotent L-groups UNillowast of Cappell [2]
References
[1] H Bass A Heller and R Swan The Whitehead group of a polynomial
extension Publ Math IHES 22 61ndash80 (1964)
[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)
[3] and J Shaneson The codimension two placement problem and
homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)
[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)
[5] P M Cohn Free rings and their relations Academic Press (1971)
[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and
Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc
18
[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)
[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)
[9] M Farber Morse-Novikov critical point theory Cohn localization and
Dirichlet units Commun Contemp Math 1 467ndash495 (1999)
[10] and A Ranicki The Morse-Novikov theory of circle-valued func-
tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)
[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)
[12] S Garoufalidis and A Kricker A rational noncommutative invariant
of boundary links httparXivmathGT0105028 (2001)
[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)
[14] A Neeman and A Ranicki Noncommutative localization and chain
complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)
[15] S P Novikov The hamiltonian formalism and a multi-valued analogue
of Morse theory Russian Math Surveys 375 1ndash56 (1982)
[16] A Pajitnov Incidence coefficients in the Novikov complex for
Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)
[17] F Quinn Open book decompositions and the bordism of automor-
phisms Topology 18 55ndash73 (1979)
[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)
[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)
19
[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)
[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)
[22] The algebraic construction of the Novikov complex of a circle-
valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)
[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111
[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)
[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-
calization J London Math Soc 64 13ndash28 (2001)
[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)
[27] L Siebenmann The obstruction to finding the boundary of an open
manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf
[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)
[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)
[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)
[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)
[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)
School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building
20
Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK
e-mail aarmathsedacuk
21
Finite domination
Codimension 1 splitting
Homology surgery theory
Codimension 2 embeddings
Open books
Boundary link cobordism
Circle-valued Morse theory
3- and 4-dimensional manifolds
The algebraic K-theory of a noncommutative localization
Matrix rings
HNN extensions
Amalgamated free products
The algebraic L-theory of a noncommutative localization
algebraic K-theory of modules and in the algebraic L-theory of quadraticforms The algebraic K-theory obstruction lives in the Whitehead groupWh(π) of the fundamental group π The L-theory obstruction lives in oneof the surgery groups Llowast(Z[π]) of Wall [32] and is defined when the topo-logical and algebraic K-theory obstructions vanish The groups Llowast(Λ) aredefined for any ring with involution Λ to be the generalized Witt groups ofnonsingular quadratic forms over Λ For manifolds of dimension gt 5 thevanishing of the algebraic obstructions is both a necessary and sufficientcondition for deforming a homotopy equivalence to a homeomorphism SeeRanicki [20] for the reduction of the Browder-Novikov-Sullivan-Wall surgerytheory to algebra
A homotopy equivalence of m-dimensional manifolds f M prime rarr M splits
along a submanifold Nn sub Mm if f is homotopic to a map (also denoted byf) such that N prime = fminus1(N) sub M prime is also a submanifold and the restrictionf | N prime rarr N is also a homotopy equivalence For codimension m minus n gt 3the splitting obstruction is just the ordinary surgery obstruction σlowast(f |) isinLm(Z[π1(N)]) For codimension m minus n = 1 2 the splitting obstructionsinvolve the interplay of the knotting properties of codimension (m minus n)submanifolds and Mayer-Vietoris-type decompositions of the algebraic K-and L-groups of Z[π1(M)] in terms of the groups of Z[π1(N)] Z[π1(MN)]
In the case m minus n = 1 π1(M) is a generalized free product ie eitheran amalgamated free product or an HNN extension by the Seifert-vanKampen theorem Codimension 1 splitting theorems and the algebraic K-and L-theory of generalized free products are a major ingredient of high-dimensional manifold topology featuring in the work of Stallings Brow-der Novikov Wall Siebenmann Farrell Hsiang Shaneson Casson Wald-hausen Cappell and the author Noncommutative localization providesa systematic development of this algebra using the intuition afforded by thetopological applications ndash see Part 2 below for a more detailed discussion
13 Homology surgery theory
For a morphism of rings with involution F Z[π] rarr Λ Cappell and Shaneson[3] considered the problem of whether a Λ-coefficient homology equivalenceof manifolds with fundamental group π is H-cobordant to a homeomor-phism Again the answer is no in general with obstructions in the topolog-ical K-theory of vector bundles and in the homology surgery groups Γlowast(F)which are generalized Witt groups of Λ-nonsingular quadratic forms over
4
Z[π] Vogel [28] [29] identified the Λ-coefficient homology surgery groupswith the ordinary L-groups of the localization Σminus1Z[π] of Z[π] inverting theset Σ of Λ-invertible square matrices over Z[π]
Γlowast(F) = Llowast(Σminus1Z[π])
and identified the relative L-groups Llowast(Z[π] rarr Σminus1Z[π]) in the localizationexact sequence
with generalized Witt groups Llowast(Z[π]Σ) of nonsingular Σminus1Z[π]Z[π])-valued quadratic linking forms on Σ-torsion Z[π]-modules of homologicaldimension 1
14 Codimension 2 embeddings
Suppose given a codimension 2 embedding Nn sub Mn+2 such as a knot orlink Let Σminus1A be the localization of A = Z[π1(MN)] inverting the set Σof matrices over A which become invertible over Z[π1(M)] By Alexanderduality the Σminus1A-coefficient homology modules
are determined by the homotopy class of the inclusion N sub M The A-coefficient homology groups Hlowast(MN A) and their Poincare duality prop-erties reflect more subtle invariants of N sub M such as knotting See Ranicki[21] for a general account of high-dimensional codimension 2 embedding the-ory including some of the applications of noncommutative localization
15 Open books
An (n + 2)-dimensional manifold Mn+2 is an open book if there exists acodimension 2 submanifold Nn sub Mn+2 such that the complement MN isa fibre bundle over S1 Every odd-dimensional manifold is an open bookQuinn [17] showed that for k gt 2 a (2k + 2)-dimensional manifold M isan open book if and only if an asymmetric form over Z[π1(M)] associatedto M represents 0 in the Witt group This obstruction was identified inRanicki [21] with an element in the L-group L2k+2(Ω
An n-dimensional micro-component boundary link is a codimension 2 embedding
Nn =⋃
micro
Sn sub Mn+2 = Sn+2
with a micro-component Seifert surface in which case the fundamental groupof the complement X = MN has a compatible surjection π1(X) rarr Fmicro
onto the free group on micro generators Duval [8] used the work of Cap-pell and Shaneson [4] and Vogel [29] to identify the cobordism group ofn-dimensional micro-component boundary links for n gt 2 with the relative L-group Ln+3(Z[Fmicro]Σ) in the localization exact sequence
with Σ the set of Z-invertible square matrices over Z[Fmicro] The even-dimensionalboundary link cobordism groups are L2lowast+1(Z[Fmicro]Σ) = 0 The cobordismclass in L2k+2(Z[Fmicro]Σ) of a (2k minus 1)-dimensional micro-component boundarylink cupmicroS
2kminus1 sub S2k+1 was identified with the Witt class of a Σminus1Z[Fmicro]Z[Fmicro]-valued nonsingular (minus1)k+1-quadratic linking form on Hk(XZ[Fmicro]) gener-alizing the Blanchfield pairing on the homology of the infinite cyclic coverof a knot The localization Σminus1Z[Fmicro] was identified by Dicks and Sontag [7]and Farber and Vogel [11] with a ring of rational functions in micro noncommut-ing variables The high odd-dimensional boundary link cobordism groupsL2lowast+2(Z[Fmicro]Σ) have been computed by Sheiham [26]
17 Circle-valued Morse theory
Novikov [15] proposed the study of the critical points of Morse functionsf M rarr S1 on compact manifolds M The lsquoNovikov complexrsquo C(Mf)over Z((z)) = Z[[z]][zminus1] has one generator for each critical point of f andthe lsquoNovikov homologyrsquo
Hlowast(C(Mf)) = Hlowast(M Z((z)))
provides lower bounds on the number of critical points of Morse functions inthe homotopy class of f generalizing the inequalities of the classical Morsetheory of real-valued functions M rarr R Suppose given a Morse functionf M rarr S1 with M = flowastR such that π1(M) = π1(M ) times Z (for the sakeof simplicity) Let Σ be the set of square matrices over Z[π1(M )][z] which
6
become invertible over Z[π1(M)] under the augmentation z 7rarr 0 There is anatural morphism from the localization to the completion
which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also
features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]
18 3- and 4-dimensional manifolds
See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds
Part 2 The algebraic K- and L-theory of general-
ized free products via noncommutative localization
A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of
the Z[π1(M)]-module chain complex C(M) of the universal cover M is the
7
assembly of an A-module chain complex constructed from the chain com-
plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24
21 The algebraic K-theory of a noncommutative localiza-
tion
Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution
0 Ps
Q T 0
such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]
was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo
TorAi (Σminus1AΣminus1A) = 0 (i gt 1)
then
(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B
(ii) the localization exact sequence extends to the higher K-groups
The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings
Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring
A =
(A1 B0 A2
)
An A-module can be written as
M =
(M1
M2
)
with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection
A1 timesA2 rarr A (a1 a2) 7rarr
(a1 00 a2
)
induces isomorphisms of algebraic K-groups
Klowast(A1)oplusKlowast(A2) sim= Klowast(A)
The columns of A are fg projective A-modules
P1 =
(A1
0
) P2 =
(BA2
)
such that
P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)
HomA(P1 P2) = B HomA(P2 P1) = 0
The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring
Σminus1A = M2(C) =
(C CC C
)
with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1
sim= Σminus1P2 The Morita equivalence
Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L
9
induces isomorphisms in algebraic K-theory
Klowast(M2(C)) sim= Klowast(C)
The composite of the functor
A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M
and the Morita equivalence is the assembly functor
A-modules rarr C-modules
M =
(M1
M2
)7rarr (C C)otimesA M
= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)
in the algebraic K-theory localization exact sequence
There are evident generalizations to k times k matrix rings for any k gt 2
23 HNN extensions
The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with
α(s)z = zβ(s) isin R lowastαβ z (s isin S)
Define the triangular 2times 2 matrix ring
A =
(R Rα oplusRβ
0 S
)
with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with
σ1 =
((1 0)0
) σ2 =
((0 1)0
) P1 =
(R0
)rarr P2 =
(Rα oplusRβ
S
)
The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies
Σminus1A = M2(R lowastαβ z)
10
Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected
M = M1 cupNtimes01 N times [0 1]
N times [0 1]
M1
By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1
Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are
injective and the universal cover M is a union
M =⋃
gisin[π1(M)π1(M1)]
gM1
11
of translates of the universal cover M1 of M1 and
g1M1 cap g2M1 =
hN if g1 cap g2z = h isin [π1(M) π1(N)]
g1M1 if g1 = g2
empty if g1 6= g2 and g1 cap g2z = empty
with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms
M zminus2M1 zminus1M1 M1 zM1 z2M1
zminus1N N zN z2N
The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps
iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)
iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)
defining a fg projective A-module chain complex
(C(M1)
C(N)
)with assembly
the cellular fg free Z[π1(M)]-module chain complex of M
coker
(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)
)
= C(M)
by the Mayer-Vietoris theorem
Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms
R rarr R lowastαβ z S rarr R lowastαβ z
A =
(R Rα oplusRβ
0 S
)rarr Σminus1A = M2(R lowastαβ z)
12
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
are just the unitary nilpotent L-groups UNillowast of Cappell [2]
References
[1] H Bass A Heller and R Swan The Whitehead group of a polynomial
extension Publ Math IHES 22 61ndash80 (1964)
[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)
[3] and J Shaneson The codimension two placement problem and
homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)
[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)
[5] P M Cohn Free rings and their relations Academic Press (1971)
[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and
Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc
18
[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)
[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)
[9] M Farber Morse-Novikov critical point theory Cohn localization and
Dirichlet units Commun Contemp Math 1 467ndash495 (1999)
[10] and A Ranicki The Morse-Novikov theory of circle-valued func-
tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)
[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)
[12] S Garoufalidis and A Kricker A rational noncommutative invariant
of boundary links httparXivmathGT0105028 (2001)
[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)
[14] A Neeman and A Ranicki Noncommutative localization and chain
complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)
[15] S P Novikov The hamiltonian formalism and a multi-valued analogue
of Morse theory Russian Math Surveys 375 1ndash56 (1982)
[16] A Pajitnov Incidence coefficients in the Novikov complex for
Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)
[17] F Quinn Open book decompositions and the bordism of automor-
phisms Topology 18 55ndash73 (1979)
[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)
[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)
19
[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)
[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)
[22] The algebraic construction of the Novikov complex of a circle-
valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)
[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111
[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)
[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-
calization J London Math Soc 64 13ndash28 (2001)
[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)
[27] L Siebenmann The obstruction to finding the boundary of an open
manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf
[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)
[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)
[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)
[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)
[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)
School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building
20
Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK
e-mail aarmathsedacuk
21
Finite domination
Codimension 1 splitting
Homology surgery theory
Codimension 2 embeddings
Open books
Boundary link cobordism
Circle-valued Morse theory
3- and 4-dimensional manifolds
The algebraic K-theory of a noncommutative localization
Matrix rings
HNN extensions
Amalgamated free products
The algebraic L-theory of a noncommutative localization
Z[π] Vogel [28] [29] identified the Λ-coefficient homology surgery groupswith the ordinary L-groups of the localization Σminus1Z[π] of Z[π] inverting theset Σ of Λ-invertible square matrices over Z[π]
Γlowast(F) = Llowast(Σminus1Z[π])
and identified the relative L-groups Llowast(Z[π] rarr Σminus1Z[π]) in the localizationexact sequence
with generalized Witt groups Llowast(Z[π]Σ) of nonsingular Σminus1Z[π]Z[π])-valued quadratic linking forms on Σ-torsion Z[π]-modules of homologicaldimension 1
14 Codimension 2 embeddings
Suppose given a codimension 2 embedding Nn sub Mn+2 such as a knot orlink Let Σminus1A be the localization of A = Z[π1(MN)] inverting the set Σof matrices over A which become invertible over Z[π1(M)] By Alexanderduality the Σminus1A-coefficient homology modules
are determined by the homotopy class of the inclusion N sub M The A-coefficient homology groups Hlowast(MN A) and their Poincare duality prop-erties reflect more subtle invariants of N sub M such as knotting See Ranicki[21] for a general account of high-dimensional codimension 2 embedding the-ory including some of the applications of noncommutative localization
15 Open books
An (n + 2)-dimensional manifold Mn+2 is an open book if there exists acodimension 2 submanifold Nn sub Mn+2 such that the complement MN isa fibre bundle over S1 Every odd-dimensional manifold is an open bookQuinn [17] showed that for k gt 2 a (2k + 2)-dimensional manifold M isan open book if and only if an asymmetric form over Z[π1(M)] associatedto M represents 0 in the Witt group This obstruction was identified inRanicki [21] with an element in the L-group L2k+2(Ω
An n-dimensional micro-component boundary link is a codimension 2 embedding
Nn =⋃
micro
Sn sub Mn+2 = Sn+2
with a micro-component Seifert surface in which case the fundamental groupof the complement X = MN has a compatible surjection π1(X) rarr Fmicro
onto the free group on micro generators Duval [8] used the work of Cap-pell and Shaneson [4] and Vogel [29] to identify the cobordism group ofn-dimensional micro-component boundary links for n gt 2 with the relative L-group Ln+3(Z[Fmicro]Σ) in the localization exact sequence
with Σ the set of Z-invertible square matrices over Z[Fmicro] The even-dimensionalboundary link cobordism groups are L2lowast+1(Z[Fmicro]Σ) = 0 The cobordismclass in L2k+2(Z[Fmicro]Σ) of a (2k minus 1)-dimensional micro-component boundarylink cupmicroS
2kminus1 sub S2k+1 was identified with the Witt class of a Σminus1Z[Fmicro]Z[Fmicro]-valued nonsingular (minus1)k+1-quadratic linking form on Hk(XZ[Fmicro]) gener-alizing the Blanchfield pairing on the homology of the infinite cyclic coverof a knot The localization Σminus1Z[Fmicro] was identified by Dicks and Sontag [7]and Farber and Vogel [11] with a ring of rational functions in micro noncommut-ing variables The high odd-dimensional boundary link cobordism groupsL2lowast+2(Z[Fmicro]Σ) have been computed by Sheiham [26]
17 Circle-valued Morse theory
Novikov [15] proposed the study of the critical points of Morse functionsf M rarr S1 on compact manifolds M The lsquoNovikov complexrsquo C(Mf)over Z((z)) = Z[[z]][zminus1] has one generator for each critical point of f andthe lsquoNovikov homologyrsquo
Hlowast(C(Mf)) = Hlowast(M Z((z)))
provides lower bounds on the number of critical points of Morse functions inthe homotopy class of f generalizing the inequalities of the classical Morsetheory of real-valued functions M rarr R Suppose given a Morse functionf M rarr S1 with M = flowastR such that π1(M) = π1(M ) times Z (for the sakeof simplicity) Let Σ be the set of square matrices over Z[π1(M )][z] which
6
become invertible over Z[π1(M)] under the augmentation z 7rarr 0 There is anatural morphism from the localization to the completion
which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also
features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]
18 3- and 4-dimensional manifolds
See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds
Part 2 The algebraic K- and L-theory of general-
ized free products via noncommutative localization
A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of
the Z[π1(M)]-module chain complex C(M) of the universal cover M is the
7
assembly of an A-module chain complex constructed from the chain com-
plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24
21 The algebraic K-theory of a noncommutative localiza-
tion
Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution
0 Ps
Q T 0
such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]
was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo
TorAi (Σminus1AΣminus1A) = 0 (i gt 1)
then
(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B
(ii) the localization exact sequence extends to the higher K-groups
The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings
Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring
A =
(A1 B0 A2
)
An A-module can be written as
M =
(M1
M2
)
with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection
A1 timesA2 rarr A (a1 a2) 7rarr
(a1 00 a2
)
induces isomorphisms of algebraic K-groups
Klowast(A1)oplusKlowast(A2) sim= Klowast(A)
The columns of A are fg projective A-modules
P1 =
(A1
0
) P2 =
(BA2
)
such that
P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)
HomA(P1 P2) = B HomA(P2 P1) = 0
The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring
Σminus1A = M2(C) =
(C CC C
)
with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1
sim= Σminus1P2 The Morita equivalence
Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L
9
induces isomorphisms in algebraic K-theory
Klowast(M2(C)) sim= Klowast(C)
The composite of the functor
A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M
and the Morita equivalence is the assembly functor
A-modules rarr C-modules
M =
(M1
M2
)7rarr (C C)otimesA M
= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)
in the algebraic K-theory localization exact sequence
There are evident generalizations to k times k matrix rings for any k gt 2
23 HNN extensions
The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with
α(s)z = zβ(s) isin R lowastαβ z (s isin S)
Define the triangular 2times 2 matrix ring
A =
(R Rα oplusRβ
0 S
)
with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with
σ1 =
((1 0)0
) σ2 =
((0 1)0
) P1 =
(R0
)rarr P2 =
(Rα oplusRβ
S
)
The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies
Σminus1A = M2(R lowastαβ z)
10
Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected
M = M1 cupNtimes01 N times [0 1]
N times [0 1]
M1
By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1
Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are
injective and the universal cover M is a union
M =⋃
gisin[π1(M)π1(M1)]
gM1
11
of translates of the universal cover M1 of M1 and
g1M1 cap g2M1 =
hN if g1 cap g2z = h isin [π1(M) π1(N)]
g1M1 if g1 = g2
empty if g1 6= g2 and g1 cap g2z = empty
with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms
M zminus2M1 zminus1M1 M1 zM1 z2M1
zminus1N N zN z2N
The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps
iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)
iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)
defining a fg projective A-module chain complex
(C(M1)
C(N)
)with assembly
the cellular fg free Z[π1(M)]-module chain complex of M
coker
(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)
)
= C(M)
by the Mayer-Vietoris theorem
Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms
R rarr R lowastαβ z S rarr R lowastαβ z
A =
(R Rα oplusRβ
0 S
)rarr Σminus1A = M2(R lowastαβ z)
12
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
are just the unitary nilpotent L-groups UNillowast of Cappell [2]
References
[1] H Bass A Heller and R Swan The Whitehead group of a polynomial
extension Publ Math IHES 22 61ndash80 (1964)
[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)
[3] and J Shaneson The codimension two placement problem and
homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)
[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)
[5] P M Cohn Free rings and their relations Academic Press (1971)
[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and
Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc
18
[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)
[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)
[9] M Farber Morse-Novikov critical point theory Cohn localization and
Dirichlet units Commun Contemp Math 1 467ndash495 (1999)
[10] and A Ranicki The Morse-Novikov theory of circle-valued func-
tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)
[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)
[12] S Garoufalidis and A Kricker A rational noncommutative invariant
of boundary links httparXivmathGT0105028 (2001)
[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)
[14] A Neeman and A Ranicki Noncommutative localization and chain
complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)
[15] S P Novikov The hamiltonian formalism and a multi-valued analogue
of Morse theory Russian Math Surveys 375 1ndash56 (1982)
[16] A Pajitnov Incidence coefficients in the Novikov complex for
Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)
[17] F Quinn Open book decompositions and the bordism of automor-
phisms Topology 18 55ndash73 (1979)
[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)
[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)
19
[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)
[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)
[22] The algebraic construction of the Novikov complex of a circle-
valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)
[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111
[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)
[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-
calization J London Math Soc 64 13ndash28 (2001)
[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)
[27] L Siebenmann The obstruction to finding the boundary of an open
manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf
[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)
[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)
[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)
[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)
[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)
School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building
20
Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK
e-mail aarmathsedacuk
21
Finite domination
Codimension 1 splitting
Homology surgery theory
Codimension 2 embeddings
Open books
Boundary link cobordism
Circle-valued Morse theory
3- and 4-dimensional manifolds
The algebraic K-theory of a noncommutative localization
Matrix rings
HNN extensions
Amalgamated free products
The algebraic L-theory of a noncommutative localization
16 Boundary link cobordism
An n-dimensional micro-component boundary link is a codimension 2 embedding
Nn =⋃
micro
Sn sub Mn+2 = Sn+2
with a micro-component Seifert surface in which case the fundamental groupof the complement X = MN has a compatible surjection π1(X) rarr Fmicro
onto the free group on micro generators Duval [8] used the work of Cap-pell and Shaneson [4] and Vogel [29] to identify the cobordism group ofn-dimensional micro-component boundary links for n gt 2 with the relative L-group Ln+3(Z[Fmicro]Σ) in the localization exact sequence
with Σ the set of Z-invertible square matrices over Z[Fmicro] The even-dimensionalboundary link cobordism groups are L2lowast+1(Z[Fmicro]Σ) = 0 The cobordismclass in L2k+2(Z[Fmicro]Σ) of a (2k minus 1)-dimensional micro-component boundarylink cupmicroS
2kminus1 sub S2k+1 was identified with the Witt class of a Σminus1Z[Fmicro]Z[Fmicro]-valued nonsingular (minus1)k+1-quadratic linking form on Hk(XZ[Fmicro]) gener-alizing the Blanchfield pairing on the homology of the infinite cyclic coverof a knot The localization Σminus1Z[Fmicro] was identified by Dicks and Sontag [7]and Farber and Vogel [11] with a ring of rational functions in micro noncommut-ing variables The high odd-dimensional boundary link cobordism groupsL2lowast+2(Z[Fmicro]Σ) have been computed by Sheiham [26]
17 Circle-valued Morse theory
Novikov [15] proposed the study of the critical points of Morse functionsf M rarr S1 on compact manifolds M The lsquoNovikov complexrsquo C(Mf)over Z((z)) = Z[[z]][zminus1] has one generator for each critical point of f andthe lsquoNovikov homologyrsquo
Hlowast(C(Mf)) = Hlowast(M Z((z)))
provides lower bounds on the number of critical points of Morse functions inthe homotopy class of f generalizing the inequalities of the classical Morsetheory of real-valued functions M rarr R Suppose given a Morse functionf M rarr S1 with M = flowastR such that π1(M) = π1(M ) times Z (for the sakeof simplicity) Let Σ be the set of square matrices over Z[π1(M )][z] which
6
become invertible over Z[π1(M)] under the augmentation z 7rarr 0 There is anatural morphism from the localization to the completion
which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also
features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]
18 3- and 4-dimensional manifolds
See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds
Part 2 The algebraic K- and L-theory of general-
ized free products via noncommutative localization
A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of
the Z[π1(M)]-module chain complex C(M) of the universal cover M is the
7
assembly of an A-module chain complex constructed from the chain com-
plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24
21 The algebraic K-theory of a noncommutative localiza-
tion
Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution
0 Ps
Q T 0
such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]
was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo
TorAi (Σminus1AΣminus1A) = 0 (i gt 1)
then
(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B
(ii) the localization exact sequence extends to the higher K-groups
The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings
Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring
A =
(A1 B0 A2
)
An A-module can be written as
M =
(M1
M2
)
with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection
A1 timesA2 rarr A (a1 a2) 7rarr
(a1 00 a2
)
induces isomorphisms of algebraic K-groups
Klowast(A1)oplusKlowast(A2) sim= Klowast(A)
The columns of A are fg projective A-modules
P1 =
(A1
0
) P2 =
(BA2
)
such that
P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)
HomA(P1 P2) = B HomA(P2 P1) = 0
The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring
Σminus1A = M2(C) =
(C CC C
)
with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1
sim= Σminus1P2 The Morita equivalence
Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L
9
induces isomorphisms in algebraic K-theory
Klowast(M2(C)) sim= Klowast(C)
The composite of the functor
A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M
and the Morita equivalence is the assembly functor
A-modules rarr C-modules
M =
(M1
M2
)7rarr (C C)otimesA M
= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)
in the algebraic K-theory localization exact sequence
There are evident generalizations to k times k matrix rings for any k gt 2
23 HNN extensions
The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with
α(s)z = zβ(s) isin R lowastαβ z (s isin S)
Define the triangular 2times 2 matrix ring
A =
(R Rα oplusRβ
0 S
)
with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with
σ1 =
((1 0)0
) σ2 =
((0 1)0
) P1 =
(R0
)rarr P2 =
(Rα oplusRβ
S
)
The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies
Σminus1A = M2(R lowastαβ z)
10
Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected
M = M1 cupNtimes01 N times [0 1]
N times [0 1]
M1
By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1
Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are
injective and the universal cover M is a union
M =⋃
gisin[π1(M)π1(M1)]
gM1
11
of translates of the universal cover M1 of M1 and
g1M1 cap g2M1 =
hN if g1 cap g2z = h isin [π1(M) π1(N)]
g1M1 if g1 = g2
empty if g1 6= g2 and g1 cap g2z = empty
with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms
M zminus2M1 zminus1M1 M1 zM1 z2M1
zminus1N N zN z2N
The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps
iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)
iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)
defining a fg projective A-module chain complex
(C(M1)
C(N)
)with assembly
the cellular fg free Z[π1(M)]-module chain complex of M
coker
(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)
)
= C(M)
by the Mayer-Vietoris theorem
Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms
R rarr R lowastαβ z S rarr R lowastαβ z
A =
(R Rα oplusRβ
0 S
)rarr Σminus1A = M2(R lowastαβ z)
12
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also
features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]
18 3- and 4-dimensional manifolds
See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds
Part 2 The algebraic K- and L-theory of general-
ized free products via noncommutative localization
A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of
the Z[π1(M)]-module chain complex C(M) of the universal cover M is the
7
assembly of an A-module chain complex constructed from the chain com-
plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24
21 The algebraic K-theory of a noncommutative localiza-
tion
Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution
0 Ps
Q T 0
such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]
was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo
TorAi (Σminus1AΣminus1A) = 0 (i gt 1)
then
(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B
(ii) the localization exact sequence extends to the higher K-groups
The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings
Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring
A =
(A1 B0 A2
)
An A-module can be written as
M =
(M1
M2
)
with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection
A1 timesA2 rarr A (a1 a2) 7rarr
(a1 00 a2
)
induces isomorphisms of algebraic K-groups
Klowast(A1)oplusKlowast(A2) sim= Klowast(A)
The columns of A are fg projective A-modules
P1 =
(A1
0
) P2 =
(BA2
)
such that
P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)
HomA(P1 P2) = B HomA(P2 P1) = 0
The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring
Σminus1A = M2(C) =
(C CC C
)
with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1
sim= Σminus1P2 The Morita equivalence
Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L
9
induces isomorphisms in algebraic K-theory
Klowast(M2(C)) sim= Klowast(C)
The composite of the functor
A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M
and the Morita equivalence is the assembly functor
A-modules rarr C-modules
M =
(M1
M2
)7rarr (C C)otimesA M
= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)
in the algebraic K-theory localization exact sequence
There are evident generalizations to k times k matrix rings for any k gt 2
23 HNN extensions
The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with
α(s)z = zβ(s) isin R lowastαβ z (s isin S)
Define the triangular 2times 2 matrix ring
A =
(R Rα oplusRβ
0 S
)
with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with
σ1 =
((1 0)0
) σ2 =
((0 1)0
) P1 =
(R0
)rarr P2 =
(Rα oplusRβ
S
)
The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies
Σminus1A = M2(R lowastαβ z)
10
Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected
M = M1 cupNtimes01 N times [0 1]
N times [0 1]
M1
By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1
Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are
injective and the universal cover M is a union
M =⋃
gisin[π1(M)π1(M1)]
gM1
11
of translates of the universal cover M1 of M1 and
g1M1 cap g2M1 =
hN if g1 cap g2z = h isin [π1(M) π1(N)]
g1M1 if g1 = g2
empty if g1 6= g2 and g1 cap g2z = empty
with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms
M zminus2M1 zminus1M1 M1 zM1 z2M1
zminus1N N zN z2N
The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps
iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)
iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)
defining a fg projective A-module chain complex
(C(M1)
C(N)
)with assembly
the cellular fg free Z[π1(M)]-module chain complex of M
coker
(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)
)
= C(M)
by the Mayer-Vietoris theorem
Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms
R rarr R lowastαβ z S rarr R lowastαβ z
A =
(R Rα oplusRβ
0 S
)rarr Σminus1A = M2(R lowastαβ z)
12
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
are just the unitary nilpotent L-groups UNillowast of Cappell [2]
References
[1] H Bass A Heller and R Swan The Whitehead group of a polynomial
extension Publ Math IHES 22 61ndash80 (1964)
[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)
[3] and J Shaneson The codimension two placement problem and
homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)
[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)
[5] P M Cohn Free rings and their relations Academic Press (1971)
[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and
Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc
18
[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)
[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)
[9] M Farber Morse-Novikov critical point theory Cohn localization and
Dirichlet units Commun Contemp Math 1 467ndash495 (1999)
[10] and A Ranicki The Morse-Novikov theory of circle-valued func-
tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)
[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)
[12] S Garoufalidis and A Kricker A rational noncommutative invariant
of boundary links httparXivmathGT0105028 (2001)
[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)
[14] A Neeman and A Ranicki Noncommutative localization and chain
complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)
[15] S P Novikov The hamiltonian formalism and a multi-valued analogue
of Morse theory Russian Math Surveys 375 1ndash56 (1982)
[16] A Pajitnov Incidence coefficients in the Novikov complex for
Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)
[17] F Quinn Open book decompositions and the bordism of automor-
phisms Topology 18 55ndash73 (1979)
[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)
[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)
19
[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)
[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)
[22] The algebraic construction of the Novikov complex of a circle-
valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)
[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111
[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)
[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-
calization J London Math Soc 64 13ndash28 (2001)
[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)
[27] L Siebenmann The obstruction to finding the boundary of an open
manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf
[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)
[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)
[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)
[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)
[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)
School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building
20
Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK
e-mail aarmathsedacuk
21
Finite domination
Codimension 1 splitting
Homology surgery theory
Codimension 2 embeddings
Open books
Boundary link cobordism
Circle-valued Morse theory
3- and 4-dimensional manifolds
The algebraic K-theory of a noncommutative localization
Matrix rings
HNN extensions
Amalgamated free products
The algebraic L-theory of a noncommutative localization
assembly of an A-module chain complex constructed from the chain com-
plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24
21 The algebraic K-theory of a noncommutative localiza-
tion
Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution
0 Ps
Q T 0
such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]
was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo
TorAi (Σminus1AΣminus1A) = 0 (i gt 1)
then
(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B
(ii) the localization exact sequence extends to the higher K-groups
The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings
Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring
A =
(A1 B0 A2
)
An A-module can be written as
M =
(M1
M2
)
with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection
A1 timesA2 rarr A (a1 a2) 7rarr
(a1 00 a2
)
induces isomorphisms of algebraic K-groups
Klowast(A1)oplusKlowast(A2) sim= Klowast(A)
The columns of A are fg projective A-modules
P1 =
(A1
0
) P2 =
(BA2
)
such that
P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)
HomA(P1 P2) = B HomA(P2 P1) = 0
The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring
Σminus1A = M2(C) =
(C CC C
)
with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1
sim= Σminus1P2 The Morita equivalence
Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L
9
induces isomorphisms in algebraic K-theory
Klowast(M2(C)) sim= Klowast(C)
The composite of the functor
A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M
and the Morita equivalence is the assembly functor
A-modules rarr C-modules
M =
(M1
M2
)7rarr (C C)otimesA M
= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)
in the algebraic K-theory localization exact sequence
There are evident generalizations to k times k matrix rings for any k gt 2
23 HNN extensions
The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with
α(s)z = zβ(s) isin R lowastαβ z (s isin S)
Define the triangular 2times 2 matrix ring
A =
(R Rα oplusRβ
0 S
)
with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with
σ1 =
((1 0)0
) σ2 =
((0 1)0
) P1 =
(R0
)rarr P2 =
(Rα oplusRβ
S
)
The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies
Σminus1A = M2(R lowastαβ z)
10
Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected
M = M1 cupNtimes01 N times [0 1]
N times [0 1]
M1
By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1
Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are
injective and the universal cover M is a union
M =⋃
gisin[π1(M)π1(M1)]
gM1
11
of translates of the universal cover M1 of M1 and
g1M1 cap g2M1 =
hN if g1 cap g2z = h isin [π1(M) π1(N)]
g1M1 if g1 = g2
empty if g1 6= g2 and g1 cap g2z = empty
with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms
M zminus2M1 zminus1M1 M1 zM1 z2M1
zminus1N N zN z2N
The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps
iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)
iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)
defining a fg projective A-module chain complex
(C(M1)
C(N)
)with assembly
the cellular fg free Z[π1(M)]-module chain complex of M
coker
(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)
)
= C(M)
by the Mayer-Vietoris theorem
Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms
R rarr R lowastαβ z S rarr R lowastαβ z
A =
(R Rα oplusRβ
0 S
)rarr Σminus1A = M2(R lowastαβ z)
12
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
are just the unitary nilpotent L-groups UNillowast of Cappell [2]
References
[1] H Bass A Heller and R Swan The Whitehead group of a polynomial
extension Publ Math IHES 22 61ndash80 (1964)
[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)
[3] and J Shaneson The codimension two placement problem and
homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)
[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)
[5] P M Cohn Free rings and their relations Academic Press (1971)
[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and
Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc
18
[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)
[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)
[9] M Farber Morse-Novikov critical point theory Cohn localization and
Dirichlet units Commun Contemp Math 1 467ndash495 (1999)
[10] and A Ranicki The Morse-Novikov theory of circle-valued func-
tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)
[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)
[12] S Garoufalidis and A Kricker A rational noncommutative invariant
of boundary links httparXivmathGT0105028 (2001)
[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)
[14] A Neeman and A Ranicki Noncommutative localization and chain
complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)
[15] S P Novikov The hamiltonian formalism and a multi-valued analogue
of Morse theory Russian Math Surveys 375 1ndash56 (1982)
[16] A Pajitnov Incidence coefficients in the Novikov complex for
Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)
[17] F Quinn Open book decompositions and the bordism of automor-
phisms Topology 18 55ndash73 (1979)
[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)
[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)
19
[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)
[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)
[22] The algebraic construction of the Novikov complex of a circle-
valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)
[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111
[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)
[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-
calization J London Math Soc 64 13ndash28 (2001)
[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)
[27] L Siebenmann The obstruction to finding the boundary of an open
manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf
[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)
[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)
[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)
[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)
[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)
School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building
20
Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK
e-mail aarmathsedacuk
21
Finite domination
Codimension 1 splitting
Homology surgery theory
Codimension 2 embeddings
Open books
Boundary link cobordism
Circle-valued Morse theory
3- and 4-dimensional manifolds
The algebraic K-theory of a noncommutative localization
Matrix rings
HNN extensions
Amalgamated free products
The algebraic L-theory of a noncommutative localization
22 Matrix rings
The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings
Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring
A =
(A1 B0 A2
)
An A-module can be written as
M =
(M1
M2
)
with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection
A1 timesA2 rarr A (a1 a2) 7rarr
(a1 00 a2
)
induces isomorphisms of algebraic K-groups
Klowast(A1)oplusKlowast(A2) sim= Klowast(A)
The columns of A are fg projective A-modules
P1 =
(A1
0
) P2 =
(BA2
)
such that
P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)
HomA(P1 P2) = B HomA(P2 P1) = 0
The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring
Σminus1A = M2(C) =
(C CC C
)
with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1
sim= Σminus1P2 The Morita equivalence
Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L
9
induces isomorphisms in algebraic K-theory
Klowast(M2(C)) sim= Klowast(C)
The composite of the functor
A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M
and the Morita equivalence is the assembly functor
A-modules rarr C-modules
M =
(M1
M2
)7rarr (C C)otimesA M
= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)
in the algebraic K-theory localization exact sequence
There are evident generalizations to k times k matrix rings for any k gt 2
23 HNN extensions
The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with
α(s)z = zβ(s) isin R lowastαβ z (s isin S)
Define the triangular 2times 2 matrix ring
A =
(R Rα oplusRβ
0 S
)
with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with
σ1 =
((1 0)0
) σ2 =
((0 1)0
) P1 =
(R0
)rarr P2 =
(Rα oplusRβ
S
)
The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies
Σminus1A = M2(R lowastαβ z)
10
Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected
M = M1 cupNtimes01 N times [0 1]
N times [0 1]
M1
By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1
Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are
injective and the universal cover M is a union
M =⋃
gisin[π1(M)π1(M1)]
gM1
11
of translates of the universal cover M1 of M1 and
g1M1 cap g2M1 =
hN if g1 cap g2z = h isin [π1(M) π1(N)]
g1M1 if g1 = g2
empty if g1 6= g2 and g1 cap g2z = empty
with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms
M zminus2M1 zminus1M1 M1 zM1 z2M1
zminus1N N zN z2N
The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps
iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)
iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)
defining a fg projective A-module chain complex
(C(M1)
C(N)
)with assembly
the cellular fg free Z[π1(M)]-module chain complex of M
coker
(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)
)
= C(M)
by the Mayer-Vietoris theorem
Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms
R rarr R lowastαβ z S rarr R lowastαβ z
A =
(R Rα oplusRβ
0 S
)rarr Σminus1A = M2(R lowastαβ z)
12
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
in the algebraic K-theory localization exact sequence
There are evident generalizations to k times k matrix rings for any k gt 2
23 HNN extensions
The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with
α(s)z = zβ(s) isin R lowastαβ z (s isin S)
Define the triangular 2times 2 matrix ring
A =
(R Rα oplusRβ
0 S
)
with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with
σ1 =
((1 0)0
) σ2 =
((0 1)0
) P1 =
(R0
)rarr P2 =
(Rα oplusRβ
S
)
The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies
Σminus1A = M2(R lowastαβ z)
10
Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected
M = M1 cupNtimes01 N times [0 1]
N times [0 1]
M1
By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1
Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are
injective and the universal cover M is a union
M =⋃
gisin[π1(M)π1(M1)]
gM1
11
of translates of the universal cover M1 of M1 and
g1M1 cap g2M1 =
hN if g1 cap g2z = h isin [π1(M) π1(N)]
g1M1 if g1 = g2
empty if g1 6= g2 and g1 cap g2z = empty
with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms
M zminus2M1 zminus1M1 M1 zM1 z2M1
zminus1N N zN z2N
The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps
iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)
iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)
defining a fg projective A-module chain complex
(C(M1)
C(N)
)with assembly
the cellular fg free Z[π1(M)]-module chain complex of M
coker
(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)
)
= C(M)
by the Mayer-Vietoris theorem
Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms
R rarr R lowastαβ z S rarr R lowastαβ z
A =
(R Rα oplusRβ
0 S
)rarr Σminus1A = M2(R lowastαβ z)
12
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
are just the unitary nilpotent L-groups UNillowast of Cappell [2]
References
[1] H Bass A Heller and R Swan The Whitehead group of a polynomial
extension Publ Math IHES 22 61ndash80 (1964)
[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)
[3] and J Shaneson The codimension two placement problem and
homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)
[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)
[5] P M Cohn Free rings and their relations Academic Press (1971)
[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and
Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc
18
[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)
[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)
[9] M Farber Morse-Novikov critical point theory Cohn localization and
Dirichlet units Commun Contemp Math 1 467ndash495 (1999)
[10] and A Ranicki The Morse-Novikov theory of circle-valued func-
tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)
[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)
[12] S Garoufalidis and A Kricker A rational noncommutative invariant
of boundary links httparXivmathGT0105028 (2001)
[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)
[14] A Neeman and A Ranicki Noncommutative localization and chain
complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)
[15] S P Novikov The hamiltonian formalism and a multi-valued analogue
of Morse theory Russian Math Surveys 375 1ndash56 (1982)
[16] A Pajitnov Incidence coefficients in the Novikov complex for
Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)
[17] F Quinn Open book decompositions and the bordism of automor-
phisms Topology 18 55ndash73 (1979)
[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)
[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)
19
[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)
[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)
[22] The algebraic construction of the Novikov complex of a circle-
valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)
[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111
[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)
[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-
calization J London Math Soc 64 13ndash28 (2001)
[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)
[27] L Siebenmann The obstruction to finding the boundary of an open
manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf
[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)
[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)
[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)
[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)
[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)
School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building
20
Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK
e-mail aarmathsedacuk
21
Finite domination
Codimension 1 splitting
Homology surgery theory
Codimension 2 embeddings
Open books
Boundary link cobordism
Circle-valued Morse theory
3- and 4-dimensional manifolds
The algebraic K-theory of a noncommutative localization
Matrix rings
HNN extensions
Amalgamated free products
The algebraic L-theory of a noncommutative localization
Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected
M = M1 cupNtimes01 N times [0 1]
N times [0 1]
M1
By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1
Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are
injective and the universal cover M is a union
M =⋃
gisin[π1(M)π1(M1)]
gM1
11
of translates of the universal cover M1 of M1 and
g1M1 cap g2M1 =
hN if g1 cap g2z = h isin [π1(M) π1(N)]
g1M1 if g1 = g2
empty if g1 6= g2 and g1 cap g2z = empty
with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms
M zminus2M1 zminus1M1 M1 zM1 z2M1
zminus1N N zN z2N
The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps
iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)
iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)
defining a fg projective A-module chain complex
(C(M1)
C(N)
)with assembly
the cellular fg free Z[π1(M)]-module chain complex of M
coker
(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)
)
= C(M)
by the Mayer-Vietoris theorem
Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms
R rarr R lowastαβ z S rarr R lowastαβ z
A =
(R Rα oplusRβ
0 S
)rarr Σminus1A = M2(R lowastαβ z)
12
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
are just the unitary nilpotent L-groups UNillowast of Cappell [2]
References
[1] H Bass A Heller and R Swan The Whitehead group of a polynomial
extension Publ Math IHES 22 61ndash80 (1964)
[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)
[3] and J Shaneson The codimension two placement problem and
homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)
[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)
[5] P M Cohn Free rings and their relations Academic Press (1971)
[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and
Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc
18
[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)
[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)
[9] M Farber Morse-Novikov critical point theory Cohn localization and
Dirichlet units Commun Contemp Math 1 467ndash495 (1999)
[10] and A Ranicki The Morse-Novikov theory of circle-valued func-
tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)
[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)
[12] S Garoufalidis and A Kricker A rational noncommutative invariant
of boundary links httparXivmathGT0105028 (2001)
[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)
[14] A Neeman and A Ranicki Noncommutative localization and chain
complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)
[15] S P Novikov The hamiltonian formalism and a multi-valued analogue
of Morse theory Russian Math Surveys 375 1ndash56 (1982)
[16] A Pajitnov Incidence coefficients in the Novikov complex for
Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)
[17] F Quinn Open book decompositions and the bordism of automor-
phisms Topology 18 55ndash73 (1979)
[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)
[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)
19
[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)
[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)
[22] The algebraic construction of the Novikov complex of a circle-
valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)
[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111
[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)
[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-
calization J London Math Soc 64 13ndash28 (2001)
[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)
[27] L Siebenmann The obstruction to finding the boundary of an open
manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf
[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)
[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)
[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)
[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)
[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)
School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building
20
Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK
e-mail aarmathsedacuk
21
Finite domination
Codimension 1 splitting
Homology surgery theory
Codimension 2 embeddings
Open books
Boundary link cobordism
Circle-valued Morse theory
3- and 4-dimensional manifolds
The algebraic K-theory of a noncommutative localization
Matrix rings
HNN extensions
Amalgamated free products
The algebraic L-theory of a noncommutative localization
of translates of the universal cover M1 of M1 and
g1M1 cap g2M1 =
hN if g1 cap g2z = h isin [π1(M) π1(N)]
g1M1 if g1 = g2
empty if g1 6= g2 and g1 cap g2z = empty
with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms
M zminus2M1 zminus1M1 M1 zM1 z2M1
zminus1N N zN z2N
The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps
iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)
iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)
defining a fg projective A-module chain complex
(C(M1)
C(N)
)with assembly
the cellular fg free Z[π1(M)]-module chain complex of M
coker
(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)
)
= C(M)
by the Mayer-Vietoris theorem
Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms
R rarr R lowastαβ z S rarr R lowastαβ z
A =
(R Rα oplusRβ
0 S
)rarr Σminus1A = M2(R lowastαβ z)
12
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
are just the unitary nilpotent L-groups UNillowast of Cappell [2]
References
[1] H Bass A Heller and R Swan The Whitehead group of a polynomial
extension Publ Math IHES 22 61ndash80 (1964)
[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)
[3] and J Shaneson The codimension two placement problem and
homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)
[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)
[5] P M Cohn Free rings and their relations Academic Press (1971)
[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and
Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc
18
[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)
[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)
[9] M Farber Morse-Novikov critical point theory Cohn localization and
Dirichlet units Commun Contemp Math 1 467ndash495 (1999)
[10] and A Ranicki The Morse-Novikov theory of circle-valued func-
tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)
[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)
[12] S Garoufalidis and A Kricker A rational noncommutative invariant
of boundary links httparXivmathGT0105028 (2001)
[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)
[14] A Neeman and A Ranicki Noncommutative localization and chain
complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)
[15] S P Novikov The hamiltonian formalism and a multi-valued analogue
of Morse theory Russian Math Surveys 375 1ndash56 (1982)
[16] A Pajitnov Incidence coefficients in the Novikov complex for
Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)
[17] F Quinn Open book decompositions and the bordism of automor-
phisms Topology 18 55ndash73 (1979)
[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)
[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)
19
[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)
[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)
[22] The algebraic construction of the Novikov complex of a circle-
valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)
[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111
[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)
[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-
calization J London Math Soc 64 13ndash28 (2001)
[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)
[27] L Siebenmann The obstruction to finding the boundary of an open
manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf
[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)
[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)
[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)
[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)
[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)
School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building
20
Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK
e-mail aarmathsedacuk
21
Finite domination
Codimension 1 splitting
Homology surgery theory
Codimension 2 embeddings
Open books
Boundary link cobordism
Circle-valued Morse theory
3- and 4-dimensional manifolds
The algebraic K-theory of a noncommutative localization
Matrix rings
HNN extensions
Amalgamated free products
The algebraic L-theory of a noncommutative localization
are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C
0 R lowastαβ z otimesS Eiαminusziβ
R lowastαβ z otimesR D C 0
with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
are just the unitary nilpotent L-groups UNillowast of Cappell [2]
References
[1] H Bass A Heller and R Swan The Whitehead group of a polynomial
extension Publ Math IHES 22 61ndash80 (1964)
[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)
[3] and J Shaneson The codimension two placement problem and
homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)
[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)
[5] P M Cohn Free rings and their relations Academic Press (1971)
[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and
Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc
18
[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)
[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)
[9] M Farber Morse-Novikov critical point theory Cohn localization and
Dirichlet units Commun Contemp Math 1 467ndash495 (1999)
[10] and A Ranicki The Morse-Novikov theory of circle-valued func-
tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)
[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)
[12] S Garoufalidis and A Kricker A rational noncommutative invariant
of boundary links httparXivmathGT0105028 (2001)
[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)
[14] A Neeman and A Ranicki Noncommutative localization and chain
complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)
[15] S P Novikov The hamiltonian formalism and a multi-valued analogue
of Morse theory Russian Math Surveys 375 1ndash56 (1982)
[16] A Pajitnov Incidence coefficients in the Novikov complex for
Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)
[17] F Quinn Open book decompositions and the bordism of automor-
phisms Topology 18 55ndash73 (1979)
[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)
[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)
19
[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)
[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)
[22] The algebraic construction of the Novikov complex of a circle-
valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)
[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111
[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)
[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-
calization J London Math Soc 64 13ndash28 (2001)
[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)
[27] L Siebenmann The obstruction to finding the boundary of an open
manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf
[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)
[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)
[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)
[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)
[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)
School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building
20
Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK
e-mail aarmathsedacuk
21
Finite domination
Codimension 1 splitting
Homology surgery theory
Codimension 2 embeddings
Open books
Boundary link cobordism
Circle-valued Morse theory
3- and 4-dimensional manifolds
The algebraic K-theory of a noncommutative localization
Matrix rings
HNN extensions
Amalgamated free products
The algebraic L-theory of a noncommutative localization
Define the triangular 3times 3 matrix ring
A =
R1 0 R1
0 R2 R2
0 0 S
and the A-module morphisms
σ1 =
100
P1 =
R1
00
rarr P3 =
R1
R2
S
σ2 =
010
P2 =
0R2
0
rarr P3 =
R1
R2
S
The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring
Σminus1A = M3(R1 lowastS R2)
(a modification of Theorem 410 of [24])
Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and
M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2
with M1M2 N connected
M1 M2N times [0 1]
By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product
π1(M) = π1(M1) lowastπ1(N) π1(M2)
14
so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms
are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C
0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0
with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]
Kn(S)oplus Niln(R1 R2 S)i1 0i2 0
Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)
25 The algebraic L-theory of a noncommutative localization
See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization
The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that
a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)
For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory
with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring
A =
(A1 B0 A2
)
17
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence
are just the unitary nilpotent L-groups UNillowast of Cappell [2]
References
[1] H Bass A Heller and R Swan The Whitehead group of a polynomial
extension Publ Math IHES 22 61ndash80 (1964)
[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)
[3] and J Shaneson The codimension two placement problem and
homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)
[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)
[5] P M Cohn Free rings and their relations Academic Press (1971)
[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and
Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc
18
[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)
[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)
[9] M Farber Morse-Novikov critical point theory Cohn localization and
Dirichlet units Commun Contemp Math 1 467ndash495 (1999)
[10] and A Ranicki The Morse-Novikov theory of circle-valued func-
tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)
[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)
[12] S Garoufalidis and A Kricker A rational noncommutative invariant
of boundary links httparXivmathGT0105028 (2001)
[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)
[14] A Neeman and A Ranicki Noncommutative localization and chain
complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)
[15] S P Novikov The hamiltonian formalism and a multi-valued analogue
of Morse theory Russian Math Surveys 375 1ndash56 (1982)
[16] A Pajitnov Incidence coefficients in the Novikov complex for
Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)
[17] F Quinn Open book decompositions and the bordism of automor-
phisms Topology 18 55ndash73 (1979)
[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)
[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)
19
[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)
[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)
[22] The algebraic construction of the Novikov complex of a circle-
valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)
[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111
[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)
[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-
calization J London Math Soc 64 13ndash28 (2001)
[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)
[27] L Siebenmann The obstruction to finding the boundary of an open
manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf
[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)
[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)
[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)
[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)
[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)
School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building
20
Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK
e-mail aarmathsedacuk
21
Finite domination
Codimension 1 splitting
Homology surgery theory
Codimension 2 embeddings
Open books
Boundary link cobordism
Circle-valued Morse theory
3- and 4-dimensional manifolds
The algebraic K-theory of a noncommutative localization
Matrix rings
HNN extensions
Amalgamated free products
The algebraic L-theory of a noncommutative localization
The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]
sending an A-module M =
(M1
M2
)to the 1-dimensional A-module chain
complex
TM TM1 =
(Mlowast
1
0
)rarr TM0 =
(B otimesA2 M
lowast2
Mlowast2
)
The quadratic L-groups of A are just the relative L-groups in the exactsequence
In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence