Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Algebraic K -Theory of Strict Ring Spectra John Rognes University of Oslo, Norway Seoul ICM 2014 John Rognes Algebraic K -Theory of Strict Ring Spectra
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Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Algebraic K -Theory of Strict Ring Spectra
John Rognes
University of Oslo, Norway
Seoul ICM 2014
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Outline
1 Algebraic K -Theory and Automorphisms of Manifolds
2 Topological Cyclic Homology and p-Complete Calculations
3 Logarithmic Ring Spectra and Localization Sequences
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Outline
1 Algebraic K -Theory and Automorphisms of Manifolds
2 Topological Cyclic Homology and p-Complete Calculations
3 Logarithmic Ring Spectra and Localization Sequences
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Symmetric Spectra (Smith)
A spectrum is a sequence of based spaces
X0,X1,X2, . . .
and maps σ : Xn ∧ S1 → Xn+1, for n ≥ 0.A symmetric spectrum is a spectrum equipped with aΣn-action on each Xn, such that
σk : Xn ∧ Sk → Xn+k
is Σn × Σk -equivariant for each n, k ≥ 0.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Symmetric Ring Spectra
The category SpΣ of symmetric spectra is closedsymmetric monoidal, with unit the sphere spectrum S andmonoidal pairing the smash product X ∧ Y .Its localization Ho(SpΣ) with respect to the stableequivalences is Boardman’s stable homotopy category.A symmetric ring spectrum is a symmetric spectrum A withassociative and unital structure maps µ : A ∧ A→ A andη : S → A.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Algebraic K -Theory of Symmetric Ring Spectra
Mandell defined K (A) as the algebraic K -theory of acategory CA of finite cell A-modules.The algebraic K -theory spectrum K (A) exhibits a groupcompletion
|hCA| → Ω∞K (A)
of the left hand classifying space, turning cofibersequences into sums.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Algebraic K -Theory of Spaces
Let X ' BG be a space, with loop group G ' ΩX .Let S[G] be the spherical group ring spectrum.Waldhausen first defined
A(X ) = K (S[G])
as the algebraic K -theory of an unstable model for thecategory of finite cell S[G]-modules, the category ofretractive spaces over X .
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
h-Cobordism Spaces
If X is a compact smooth manifold, let H(X ) be the spaceof h-cobordisms (W ; X ,Y ) with X at one end:
∂W = X ∪ Y , X '→W '← Y
Let H (X ) = colimk H(X × [0,1]k ) be the stableh-cobordism space.
Theorem (Igusa)
H(X )→H (X ) is about n/3-connected, for n = dim X.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
The Stable Parametrized h-Cobordism Theorem
A(X ) = K (S[G]) splits as
A(X ) ' S[X ] ∨Wh(X ) ,
defining the Whitehead spectrum.Let ΩWh(X ) = Ω∞+1Wh(X ) be the looped Whiteheadspace.
Theorem (Waldhausen–Jahren–R.)
There is a natural homotopy equivalence H (X ) ' ΩWh(X ).
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Diffeomorphism Groups: Rational
When X is contractible, A(∗) = K (S) ' S ∨Wh(∗).
Theorem (Borel)
Ki(S)⊗Q ∼= Ki(Z)⊗Q ∼=
Q for i = 0 or 4k + 1 6= 1,0 otherwise.
Example (Farrell–Hsiang)
πiDiff (Dn rel ∂Dn)⊗Q ∼=
Q for i = 4k − 1, n odd,0 otherwise,
for i up to about n/3.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Outline
1 Algebraic K -Theory and Automorphisms of Manifolds
2 Topological Cyclic Homology and p-Complete Calculations
3 Logarithmic Ring Spectra and Localization Sequences
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Topological Cyclic Homology
Bökstedt–Hsiang–Madsen constructed a naturalcyclotomic trace map
K (A)→ TC(A; p)
to the topological cyclic homology of A.It is a homotopy limit
TC(A; p) = holimn,R,F
THH(A)Cpn
of cyclic fixed points of the topological Hochschildhomology of A.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Nilpotent extensions
An integral version satisfies TC(A)∧p ' TC(A; p)∧p .
Theorem (Dundas–Goodwillie–McCarthy)Let A→ B be a map of connective symmetric ring spectra, withπ0(A)→ π0(B) surjective with nilpotent kernel. The square
K (A) //
K (B)
TC(A) // TC(B)
is homotopy Cartesian.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
The Sphere Spectrum and the Integers
ExampleHomotopy Cartesian square
K (S)∧p //
K (Z)∧p
TC(S; p)∧p // TC(Z; p)∧p .
R. used this to calculate H∗ and π∗ of
K (S)∧p ' S∧p ∨Wh(∗)∧p
for regular primes p.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
K -Theory of the Sphere Spectrum: Cohomology
Let A be the mod p Steenrod algebra.For p = 2 let C ⊂ A be generated by admissible SqI
where I = (i1, . . . , in) with n ≥ 2 or I = (i) with i odd.
Theorem (R.)
The mod 2 cohomology of Wh(∗) is the nontrivial extension
Σ−2C/A (Sq1,Sq3)→ H∗Wh(∗)→ Σ3A /A (Sq1,Sq2)
of A -modules.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
K -Theory of the Sphere Spectrum: Homotopy
Example (R.)
The homotopy groups of Wh(∗), modulo p-torsion for irregularprimes p, begin:
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Localization and Descent for Algebraic K -Theory
Seek a conceptual understanding of these calculationalresults on K (A)p for A = S.Can we recover K (A)p from K (B)p for suitably localsymmetric ring spectra B?Can we descend to K (B)p from K (C)p for appropriateextensions B → C?Is there a simple description of K (Ω)p for sufficiently largesuch extensions B → Ω?
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Algebraic K -Theory of Topological K -Theory
Adams summand A = `p of kup, with π∗`p = Zp[v1].
Localization B = Lp, with π∗Lp = Zp[v±11 ].
Lp // KUp
Sp // `p
OO
φ // kup
OO
// HZp
Theorem (Blumberg-Mandell)
Homotopy cofiber sequence
K (`p)→ K (Lp)→ ΣK (Zp) .
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Chromatic Redshift
For p ≥ 5, the type 2 Smith–Toda complex
V (1) = S ∪p e1 ∪α1 e2p−1 ∪p e2p
is a ring spectrum up to homotopy, with v2 ∈ π2p2−2V (1).
Theorem (Ausoni–R.)
V (1)∗K (`p) and V (1)∗K (Lp)
are finitely generated free Fp[v2]-modules, each on 4p + 4generators, up to small error terms.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Lichtenbaum–Quillen Conjecture
Suggests that K (Ω)p is a connective form of the Lubin–Tatespectrum E2, with π∗E2 = WFp2 [[u1]][u±1] andV (1)∗E2 = Fp2 [u±1].
Conjecture (R.)For purely v1-periodic commutative symmetric ring spectra Bthere is a spectral sequence
E2s,t = H−s
mot(B;Fp2(t/2)) =⇒ V (1)s+tK (B) .
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
E2-Term for V (1)∗K (Lp), p = 5
••
2p2
•
•
•
|
•
•
|
•
•
|
•
•••
••
2p
•
•
•
•
•
•••
•
0
−3
−2
−1
0
s/t
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Beilinson–Lichtenbaum Conjecture
SetH r
et(Lp;Fp2(∗)) = v−12 H r
mot(Lp;Fp2(∗)) .
Observe motivic truncation:
H rmot(Lp;Fp2(m)) ∼=
H r
et(Lp;Fp2(m)) for 0 ≤ r ≤ m,0 otherwise.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Tate–Poitou Duality
Symmetry about (s, t) = (−3/2,p + 1) similar to arithmeticduality.
Conjecture (R.)For finite extensions B of Lp there is a perfect pairing
H ret(B;Fp2(m))⊗ H3−r
et (B;Fp2(p+1−m))
∪→ H3et(B;Fp2(p+1)) ∼= Z/p
for each r and m.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Outline
1 Algebraic K -Theory and Automorphisms of Manifolds
2 Topological Cyclic Homology and p-Complete Calculations
3 Logarithmic Ring Spectra and Localization Sequences
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Logarithmic Geometry
Seek to realize more of motivic cohomology as Galoiscohomology.Difficult to classify/construct ramified extensions B → C byobstruction theory.Tamely ramified extensions behave as unramified whenrigidified by logarithmic structures.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Logarithmic Rings (Fontaine–Illusie, Kato)
A pre-log ring consists ofa commutative ring R;a commutative monoid M;a monoid homomorphism α : M → (R, ·).
Log ring if α−1GL1(R)→ GL1(R) is an isomorphism.Trivial log structure on R has M = GL1(R).Localization R → R[M−1] factors in log rings as
R → (R,M)→ R[M−1] .
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
J -spaces (R., Sagave, Schlichtkrull)
The “underlying graded space” of a symmetric spectrum Ais a J -shaped diagram of spaces
ΩJ (A) : (n1,n2) 7→ Ωn2An1
Indexing category J is isomorphic to Quillen’s constructionΣ−1Σ, with BJ ' QS0.Homotopy type of a J -space X : J → S is detected byXhJ = hocolimJ X . Positive projective model structure.Convolution product X Y maps to smash product underSJ [−] : SJ → SpΣ, Quillen adjoint to ΩJ (−) : SpΣ → SJ .
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Topological logarithmic structures
DefinitionA pre-log ring spectrum consists of
a commutative symmetric ring spectrum A;a commutative J -space monoid M;a commutative J -space monoid map α : M → ΩJ (A).
Log ring spectrum if α−1GLJ1 (A)→ GLJ1 (A) isJ -equivalence.Trivial log structure on A has M = GLJ1 (A) ⊂ ΩJ (A).Localization A→ A[M−1] = A ∧SJ [M] SJ [Mgp] factors as
A→ (A,M)→ A[M−1] .
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
The replete bar construction
The group completion η : M → Mgp makes (Mgp)hJ agroup completion of the E∞ space MhJ .The cyclic bar construction Bcy(M) is the usual simplicialobject [q] 7→ M M · · · M.The replete bar construction is a homotopy pullback
Bcy(M)ρ //
Brep(M) //
Bcy(Mgp)
M = // M // Mgp
Repletion in topology plays the role of working with fineand saturated log structures in algebra.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Logarithmic Topological Hochschild Homology
DefinitionLog THH of a pre-log ring spectrum (A,M, α) is the pushout
SJ [Bcy(M)]ρ //
SJ [Brep(M)]
THH(A)
ρ // THH(A,M)
of cyclic commutative symmetric ring spectra.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Log Étale Extensions
(A,M)→ (B,N) is formally log étale ifB ∧A THH(A,M) ' THH(B,N).The direct image log structure of (B,N) along j : A→ B isj∗N = ΩJ (A)×ΩJ (B) N.
Theorem (R.–Sagave–Schlichtkrull)
φ : (`p, j∗GLJ1 (Lp))→ (kup, j∗GLJ1 (KUp))
is log étale.
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Localization Sequences
Theorem (R.–Sagave–Schlichtkrull)Let E be a d-periodic commutative symmetric ring spectrum,with connective cover j : e→ E. Homotopy cofiber sequence
THH(e)ρ→ THH(e, j∗GLJ1 (E))
∂→ ΣTHH(e[0,d))
where e[0,d) is the (d − 1)-th Postnikov section of e.
ExampleHomotopy cofiber sequence
THH(`p)→ THH(`p, j∗GLJ1 (Lp))→ ΣTHH(Zp) .
John Rognes Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Future Work
Develop log TC, with a cyclotomic trace map from logK -theory, related to K (A[M−1]).Develop log obstruction theory to realize tamely ramifiedextensions A→ B as part of log étale extensions(A,M)→ (B,N).
John Rognes Algebraic K -Theory of Strict Ring Spectra