The Homotopy Type of a 4-Manifold with finite ~"Sandamental Group by Stefan Bauer* ABSTRACT: ... is determined by its quadratic 2-type, if the 2-Sylow subgroup has 4-periodic cohomology. The homotopy type of simply connected 4-manifolds is determined by the intersection form. This is a well-known result of J.H.C. Whitehead and 3. Mitnor. In the non-simply connected case the homotopy groups ~rl and 7~ and the first k-invariant k E H3(71, v2) give other homotopy invariants. The quadratic 2-type of an oriented closed 4-manifold is the isometry class of the quadruple [71(M), ~r2(M), k(M),~/(~)], where 7(It:/) denotes ~ the intersection form on 72(M) = H2(M). An isometry of two such quadruples is an isomorphism of 71 and 72 which induces an isometry on 7 and respects the k-invariant. Recently [H- K] I. Hambleton and M. Kreck, studying the homeomorphism types of 4-manifolds, showed that for groups with periodic cohomology of period 4 the quadratic 2-type determines the homotopy type. This result can be improved away from the prime 2. Theorem: Suppose the 2-Sylow subgroup of G has 4-periodic cohomology. Then the homotopy type of an oriented 4-dimensional Poincarfi complex with fundamental group G is determined by its quadratic 2-type. I am indebted to Richard Swan for showing me proposition 6. Furthermore I am grateful to the department of mathematics at the University of Chicago for its hospitality during the last year. * Supported by the DFG
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The Homotopy Type of a 4-Manifold
with finite ~"Sandamental Group
by Stefan Bauer*
A B S T R A C T : ... is determined by its quadratic 2-type, if the 2-Sylow subgroup has 4-periodic cohomology.
The homotopy type of simply connected 4-manifolds is determined by the intersection form. This is a well-known result of J.H.C. Whitehead and 3. Mitnor. In the non-simply connected case the homotopy groups ~rl and 7~ and the first k-invariant k E H3(71, v2) give other homotopy invariants. The quadrat ic 2-type of an oriented closed 4-manifold is the isometry class of the quadruple [71(M), ~r2(M), k(M),~/(~)], where 7(It:/) denotes
~
the intersection form on 72(M) = H2(M). An isometry of two such quadruples is an isomorphism of 71 and 72 which induces an isometry on 7 and respects the k-invariant.
Recently [ H - K] I. Hambleton and M. Kreck, studying the homeomorphism types of 4-manifolds, showed that for groups with periodic cohomology of period 4 the quadratic 2-type determines the homotopy type.
This result can be improved away from the prime 2.
Theorem: Suppose the 2-Sylow subgroup of G has 4-periodic cohomology. Then the homotopy type of an oriented 4-dimensional Poincarfi complex with fundamental group G is determined by its quadratic 2-type.
I am indebted to Richard Swan for showing me proposition 6. Furthermore I am grateful to the department of mathematics at the University of Chicago for its hospitality during the last year.
* Supported by the DFG
Let X be an oriented 4-dimensional Poincar6 complex with finite fundamental group, f : X --~ B its 2-stage Postnikov approximation, determined by ~rl, 7r2, and k, a.nd let ~,(X) denote the intersection form on /72(2). Then S PD (B, 7(X)) denotes the set of homotopy types of 4-dimensional Poincar~ complexes Y, together with 3-equivalences g : Y ---* B, such that f and g induce an isometry of the quadratic 2-types. The universal cover /) is an Eilenberg-MacLane space and hence, by [MacL], H4(/)) P(~r2(B)), the ZTh(B)-modute F(Tr2(B)) being the module of symmetric 2-tensors, i.e. the kernel of the map (1 - T): ~r 2 (B) ® 7r2 (B) -* 7c~ (B) ® 7r2 (B), (1 - 7-)(a ® b) = a ® b - b ® a. The intersection form on 2 corresponds to L[2] of the fundamental class [21 • Hal2; z) . Le t / : / . denote Tate homology.
P r o p o s i t i o n 1: If X is a Poincarfi space with finite fundamental group G, then there is a. bijection [Io( G; ~ra( X) ) , , sPD ( B, "y( X)).
The proof uses a lemma of [H-K]:
L e m m a 2: Let, ( X , f ) and ( Y , g ) b e elements in sPD(B, 7(X)). Then the only obstruction for the existence of a homotopy equivalence h : X ~ Y over B is the vanishing ofg.[Y] - f . [X] • H4(B).
L e m m a 3: Given a diagram
Z ~ M
Z
such that the torsion in the cokernel of ~ is annihilated by n, then the torsion subgroup in the pushout K is isomorphic to the torsion subgroup of coker(c~).
P r o o f of 3: Since the torsion subgroup of M maps injectively into K as well as into coker(a), we may assume it trivial. Then M is isomorphic to NO < z > with a(1) = mz for an integer m dividing n. The pushout then is isomorphic to ( N @ Z @ Z ) / < (0, m, n) > ~ M • Z/m. &
P r o o f of p r o p o s i t i o n 1: Let (X, f ) and (Y, g) be elements in SPD(B) such that f and g induce an isometry of the quadratic 2-types. Let 7(X) = "y(Y) = "y denote the inter- section form on H2(X ~) and H2(]'z). By [W] one has ~r3(X) - F(~r2(X))/(7) -- H4(/~,)~ ~)
and ~ra(X) ®zv Z ~ Ha(B, X). In the pushout diagramm:
0 0 0
o ~ Ha(2) ® z a Z ~ Ha(/)) ® ~ Z , Ha(/), 2 ) ® z c Z ¢ 1 1 1 ~-
0 ~ H4(X) , Ha(B) .... ~ Ha(B, X) J. .~ +
0 ~ Ha(X, 2) ~-; Ha(B,~)) , 0
0 0
, 0
0
the torsion subgroup of H4(B, X) is isomorphic to the torsion subgroup of H4(B) by lemma 3: The module Ha(/), 2 ) is torsion free. Hence the torsion subgroup of Ha(/), 2 ) ®zc Z is annihilated by the order n of the group G. Note that ¢ is just multiplication by n. In particular one has
Since X and Y have the same quadratic 2-type, • [X] = ~.[1~], hence we have L [ X ] - g.[~'] • To~,io,~(HaB). This gives an injection
sPD( B, 7) ~-~/:/o(G;Trs(X)).
What about surjectivity? Let K C A" denote a subspace, where one single orbit is deleted. Let a e ~ra(K) map via the surjection ~r3(K) --* 7ra(X) --* ~ra(X) ® z c Z to a given element & 6 f/o(G;~rs(X)). Let f¢ be the image of 1 6 ZG ~ tIa(f(, K) ~- 7r4(2, K) '--* 7ra(K). Now let k : S 3 ---* K represent c~+fl and define X~ := ( K U k ( G x Da))/G. One has to show that X~ is an orientable Poincar4 space. Orientability is clear, since Ha(X~) ~- Ha(X~,K) ~- Z. Let f : Xa --* B extend fIK/a. The intersection form on ) ~ is determined by
.~.[Xc,] - trf(f,~.[X~]) e Ha(2). But we have fa.[Xa] = f ,[X] + o~: In the following diagram 1 6 Z ~ 7ra(X, B) is mapped to f ,[X] e Ha(B).
H4(X) ~ H4(X,K/G) ~,-- Ha(X,K) '~ m ( X , K ) - - , ~ra(K) f t l 1 l l=
H~(B) = ~ H~(B, K / a ) , - -~ H~([3, K) = ~ ~ ( B , K) ~- ,~ (K)
If the upper row is replaced by the corresponding row for X~ and the vertical maps by the ones induced by f~, then t E ZG is mapped (counterclockwise) to f~.[X~] on the one hand, on the other hand (clockwise) to f . [X] + o~. Since the torsion element o~ lies in the kernel of the transfer, one immediately gets /o.[x~] = f.[x].
In the sequel all ZG-modules have underlying a free abelian group.
The short exact sequence
0 ~ Z ~-L r ( ~ X ) , ~3(X) --* 0
gives rise to an exact sequence in Tate homology:
H 0 ( a ; Z ) , H 0 ( C ; r ( ~ X ) ) , H0(a ;~3(X) ) ~ H _ I ( C ; Z ) ~, H _ ~ ( a ; r ( ~ x ) )
Here/: /0(G; Z) = 0 and / : /_I(G;Z) ~ Z / I Gl .The sequence above gives the connection to [H-K], theorem(]. l) .
In order to analyze this sequence, I recall some facts from [H-K],§g2 and 3.
Facts: 1)
2)
3)
F ( Z G ) = (~i Z[G/Hi] ® F, ,,,here the summation is over all subgroups Hi of order 2 and F is a free ZG-module. r(zG) r([) • z a r(z*) • ZG.Here I denotes the augmentation ideal, I* its dual. The modules D3Z and SaZ are (stably!) defined by exact sequences
O--~QsZ-~ F2-~ FI..-~ Fo--~ Z--+O and
O--+ Z ~ FI ~ F2 ~ F3 ~ S3Z ~ O with free modules F,. There is an exact sequence
0 ~ f~az , ~r~(X) • r Z G ~ S3Z ~ 0
L e m m a 4: If 0 ---* A --~ B --* C ~ 0 is a short exact sequence of ZG-modules, which are free over Z, then there are short exact sequences
0 ~ P ( A ) , F(B) , , , , D ~ 0 and
0 ~ A ® z C , D - - - * F ( C ) ~ 0 .
P r o o f i Given Z-bases {ai}, {c/} and {ai,~j} of A, C and B, the map h : a, @c/ ---* ai ® ci + cJ ® ai is well-defined and equivariant modulo F(A). &
To prove the theorem, it suffices to show that H0(G; 7r3(X)) = 0. This in turn can be done separately for each p-Sylow subgroup Gp of G.
P r o p o s i t i o n 5: The map 7. : H - l ( G p ; Z ) ~ / - / - l (@; F(,r2(X))) is injective, if either p is odd or r e sg 7r2(X) ~ A @ B splits such that the rank of B over Z is odd. In general the kernel is at most of order 2.
Proof i For the sake of brevity, let 7r denote 7r2(X) and also let F denote the module r(Tr). Now look at the following sequence of maps:
a *~ t race ¢ : Z "Y, P ~ ~r @ ~r ~ gom(Tr', ~r), gom(~r, ~r) Z.
A genera.tor of Z is mapped in Hom(Tr*, ~) to the Poincar6 map a : 7r* -- H2(2) --~ H : ( 2 ) = rr, and then to the element id e Horn(or, 7:). So we have ¢(1) = rankz(Tr).
Fact 3) gives rankz(Tr) =- - 2 mod I G h hence the i~duced selfmap ¢ , of Z/I@I af-/_ ~(@; Z) is multiplication by -2. This proves, that the kernel is at most of order 2. In particular it is trivial, if p is odd.
In case p = 2 and resaa rr -~ A @ B, such that the rank of the underlying group
of B is odd, one can replace the map IIom(rr, 7r) t2-~ ~ Z by the map Hom(rr, Tr) , . . i*
Horn(B, B) tra,¢ Z in the defining sequence for ¢. A similar argument as above for p odd gives the claim. &
R e m a r k : The module resa G ~r~(X) ahvays splits, if H4(G;Z) ~ Ext~a(SaZ,fl3Z) has no 2-torsion, in particular if ~2 has 4-periodic cohomology.
P r o p o s i t i o n 6: Let A denote either ~'~Z or S'~Z and let r be the selfmap of A ® A which permutes the factors. Then (-1)n~ - induces the identity on [ /0(G;A N A).
P roof i Let F. --~ Z be a free resolution of Z and let _F. be the truncated complex with Fi = Fi for i < n - 1, ~',, = f/'~ and/~ = 0 else. There is an obvious projection f : F. --~/~'.,
such that fn = 0n. The tensor product F.®F. = F. ~ again is a free resolution of Z and/~.2 is a truncatedfree resolution of Z with/v~,~ = f~Z @ OZ. The chain map f N f induces an
isomorphism of H. (/,-'.2 @z a Z) and - 2 H.(F. @zaZ) in the dimensions * _< 2n. The selfmap of F~., as usual defined by t (z ® y) = (--1)deg(x)deg(y)x ® y, is a chain automorphism,
inducing the identity on the augmentation, hence on all derived functors, in particular on /~.2. H . (/7. 2 ® z c Z) = H. (G; Z). In the same way an involution t can be defined on and
f ® f commutes with t. Obviously ~;,, = (--1)'~r. Hence (-1)'%" induces the identity on
® z c z ) = ::o(a;z). The proof for S '~ Z is dual. &
P r o o f o f t h e t h e o r e m : By proposition 1, it suffices to show that / t0(G;za(X)) vanishes. By proposition 4 and the remark following it, this group is isomorphic to /;/0(G; F(Tr~ (X))). In order to show that this group vanishes it suffices, by lemma 3, to show that [to(G;A) vanishes for A e {[ ' (aaZ),F(SaZ),f23Z ® SaZ) But / I0 (G;aaZ ® SaZ) - f /0(G;Z) = 0. Given a module B (with underlying free abelian group), there is a short exact sequence
0 .--, F(B) - - , B ® B . . , A2(B) ---+ 0.
The map % which flips the both factors, induces, if applied to B E {f~aZ,S3Z} the following diagram:
--* / ) I (G;A(B)) ~ !flo(G;r(B)) ~ H o ( G ; B ® B ) -* I ( - i d ) I ie ~ ( - i d )
The right vertical map is ( - i d ) by proposition 5. This diagram shows that any element in H0(G;I'(B)) is annihilated by 4.In particular this group vanishes, if G is a p-group for an odd prime p. That //0(G2; F(B)) vanishes, if G~ has 4-periodic cohomology, follows at once from the facts 1 - 3, since in this case f'taZ = I* @ n Z G and SaZ = I @ n Z G
Final R e m a r k : An elementary but lengthy computation shows F(SaZ) - Z/2e Z/2 and F(f~3Z) = 0 for G = Z/2eZ/2. In particular the group fIo(Z/2®Z/2;F(~3Z®S3Z)) is nontrivial. Hence the argument above won't work in general.
REFERENCES
[B 1]
[H-K]
[MacL]
[w]
K.S. Brown: CohomoIogy of groups. GTM 87, Springer-Verlag, N.Y. 1982 R. Brown: Elements of Modern Topology. McGraw- Hill, London, 1968 I. Hambleton and M. Kreck: On the Classification of Topological 4-Manifolds with finite Fundamental Group. Preprint, 1986 S. MacLane: Cohomology theory of abelian groups. Proc. Int. Math. Congress, vol. 2 (1950), pp 8 - 14 J.H.C. Whitehead: On simply connected 4-dimensionM polyhedra. Comment. Math. Helv., 22 (1949), pp .18 - 92.
the two equivalence classes of lines over (1,1,0,0), giving I
orbit here. Then adding the number of orbits over each of the 15
lines at level 2 gives 40 orbits for A with A/F generated by
PI' giving the second line of the table.
From this point on the computation is routine.
43
proposition 2.2. For the following untwisted groups A, the
number of equivalence classes of anisotropic pairs at level A is
as stated:
[A: F] Generators of A/F ............. Number of equivalence classes
1 - 52
2 Pl 36
4 PI' P2 26
4 PI' ql 28
4 PI' q2 25
8 PI' P2' P3 22
8 Pl' P2' ql 20
8 PI' P2' q3 18
64 PI' P2' P3' q1' q2' q3 I0
Proof. This is entirely analogous to the proof of the preceding
proposition (only slightly more complicated as there are more
choices of representatives). By theorem 3.7 of [LW3] we know all
the anisotropic pairs of level F. (Each of the I0 anistropic
pairs at level 2 is covered by 16 at level 4, and by either 4 or 16
at level F. There are a total of 52 of these at level f.) For
example, the pair A = {6,6±} at level 2, with
= (i,I,0,0)^(0,0,I,0), is covered by the four equivalence
classes of pairs at level F whose representatives we may take
to be ~ and its orthogonal complement ~±, with
= (l,l,O,O)A(O,O,l,O), (l,l,O,O)A(O,O,l,2),
(l,l,0,0)A(0,2,1,0), (1,1,0,0)^(0,2,1,2) mod 4.
Then Pl acting on the first one of these planes sends it to
(1,-1,2,0)^(0,0,1,0). But adding twice the second vector to the
first, we see that this is the same plane as (l,-l,0,0)A(0,O,l,0).
Also, in the proof of 2.1, we observed that the line (1,-1,0,0)
is equivalent to (i,I,0,0), so we conclude that this plane is
equivalent to (i,I,0,0)^(0,0,I,0), i.e. Pl acts trivially on
(1,1,0,0)A(0,0,1,0).
Otherwise the computation is straightforward.
44
3. The action on excised components.
We see from 1.16 that we must count the number of components
of ZA' i.e. the number of orbits of components of Z = ZF under
the action of A/F.
Proposltion 3.!. For each untwisted subgroup A of F(2),
number of irreducible components of A A is as follows:
the
[A: P] Generators of A/P .............. Number of cgmponents
1 - 3 0
2 Pl 24
4 Pl ' P2 20
4 Pl ' ql 21
4 Pl ' q2 19
8 Pl ' P2' P3 18
8 Pl ' P2' ql 17
8 Pl ' P2' q3 16
64 Pl ' P2' P3' q l ' q2' q3 12
Proof. Again there are many cases and we shall merely indicate a
few.
Recall f: X--+ X0 is a branched covering with group
(Z/2) + (Z/2) generated by p and q, with each of f-l(0),
f-l(1), and f-l(=) having cardinality two. The following sche-
matic represents this cover:
I I
2. Z.
3
+ ~.
Q Q • o | ore,
45
Let X (resp. X ) denote the quotient of X by the action of p P q
(resp. of q). Then Xp (resp. Xq) is a 2-fold branched cover of
X0, branched over 1 and ~ (resp. 0 and ~). We continue to let
f denote the covering projection. Then f-l(0), f-l(1), f-l(~)
have cardinallty 2, I, I in Xp (resp. cardinality i, 2, i in X2).
An irreducible component of AA projects onto one of the
following types of components in A0: (*,x,y), (x,*,y), (x,y,*),
(x,x,y), (x,y,x), or (y,x,x), where * = 0, I, or ~ and x
and y are arbitrary. Thus we have six kinds of components, and
we will gather the number of each kind into a 6-tuple, whose sum is
the number of components of AA" For example, when A = F(2),
AA = A0 has the 6-tuple (3,3,3,1,1,1,) and so has 12 components.
a) The case A = F. The 6-tuple is (6,6,6,4,4,4,), as
follows: Here the covering space is X × X x X. F-l(*,x,y) has 6
components as f-l(o) u f-l(1) u f-l(~) has cardinality 6, giving
the first entry, and the second and third are identical. The
fourth entry is the number of components of X , the inverse of
the diagonal in X x X. But this inverse consists of {(Xl,X 2) I x 1
and x 2 differ by a covering translation}, and so has 4 components
(as the group of covering translations has 4 elements), and the
fifth and sixth entries are identical.
b) The case A/F generated by PI" Now the 6-tuple is
(4,6,6,2,2,4). Here the covering space is X × X x X. F-l(*,x,y) P
has 4 components as f-l(0) u f-I u f-l(~) has cardinality 4 in
giving the first entry. The second and third are as in a). P,
The fourth and fifth entries are the number of components of the
inverse image of the diagonal in X x X. But this inverse image P
is the quotient of X (as in a)) by the group generated by
p x id: X x X--+ X x X, and this quotient has two components. The
sixth entry is as in a).
e) The case A/F generated by Pl and P2" The 6-tuple is
× X x X. In par- (4,4,6,2,2,2) and the covering space is Xp P
tlcular note that the fourth entry is 2 by the same argument as
in a) •
46
d~ The case A/r generated by Pl and ql"
is (3,6,6,1,1,4). Here the covering space is
the argument is similar to b).
Now the 6-tuple
X0 x X x X, and
e~ The case A/F generated by Pl and q2" The 6-tuple is
(4,4,6,1,2,2) and the covering space is X x X x X. The only P q
(subtle) difference between this and case c) is the following: The
inverse image of the diagonal of X0 x X0 in X x X has 4 com-
ponents. Under the action of the group generated by Pl and
P2 they are identified to two components in X x X (i.e. this P P
group acts on the 4 components with Pl and P2 each acting non-
trivially but giving the same identification), while here, under
the action of the group generated by Pl and q2 they are identi-
fied to one component in X x X (i.e. this group acts on the 4 P q
components with Pl and P2 each acting non-trivlally but giving
different identifications).
The remaining cases are similar.
47
4 . The r e p r e s e n t a t i o n o f G on the homology o f M r .
In this section we obtain our main result. We use the
c a l c u l a t i o n s of s e c t i o n 2 and 3, which give dim H4(M A) fo r some
A, to decompose H4(M F) into a sum of irreducible representations
of G = r(2)/r.
First we assemble some information.
Prpposlt~gn 4.1. For the following untwisted subgroups
F c A c F(2), dim H4(MA: ~) is as stated:
[A: F] .................... Generators of A/F ......... Dimensio ~
1 - 79
2 Pl 55
4 PI' P2 41
4 PI' ql 43
4 PI' q2 38
8 PI' P2' P3 35
8 PI' P2' ql 31
8 PI' P2' q3 28
64 PI' P2' P3' ql' q2' q3 16
Proof. Immediate from 0.4, 2.1, 2.2, 1.16 and 3.1.
Now G has 64 irreducible representations, all I-dimensional,
which are obtained by letting each of the six generators PI' P2'
P3' ql' q2' q3 act by multiplication by ±i. We will denote an
irreducible representation of G by e = (el,...,e 6) where each
e i is + or - according as the corresponding generator acts by
+I or -I.
Let R = H4(M F) regarded as a representation space of G.
Let R(e) be the subspace on which G acts by the representation
e. Our problem is to determine dim R(e) = the multiplicity of e
in R. The answer is this:
1"neorew 4.2. The multiplicities of the irreducible representations
of G in its action on H4(M F) are given by the fo l lowing t ab le :
48
e l , e 2 , e 3
e 4 , e s , e 6
+++
+÷--
÷--+
--÷÷
÷++ +÷-- ÷--+ --÷+ +---- --+-- ---+ ----
16 3 3 3 3 3 3 1
3 3 0 0 0 0 0 0
3 0 3 0 0 0 0 0
3 0 0 3 0 0 0 0
3 0 0 0 3 1 1 0
3 0 0 0 1 3 1 0
3 0 0 0 1 1 3 0
1 0 0 0 0 0 0 1
(Thus for example, the multiplicity of the representation where
pl,P2,P3 (resp. ql,q2,q3 ) act by (+I,-I,-I) (resp. (-I,+i,-i))
is the intersection of the column labelled +-- and the row
labelled -+- and is I. (Note in this representation (rl,r2,r 3)
act by (-I,-I,+I).)
Remark 4.3. The reader will observe that the multiplicity of each
non-trivial representation is one less than a power of two. Why
this should be so, or what it means, is a complete mystery to us.
Gathering the irreducible representation of G into A(G)-
equivalence classes, we may rephrase the theorem as follows. (Note
that when we compare representations of different Vi's, we are
using their identification to V.)
Theorem 4.4. As a representation space of
R = H4(M F) decomposes as f o l l o w s :
G =V 1 xV 2 xV 3,
Type of irreducible No. of irreducibles Multiplicity Total of this type of each in R Dimension
in R
Trivial 1 16 16
V i acts non-trlvially 9 3 27
for one value of i
49
V i acts non-trivially
for two values of i - both act the same way
V i acts non-trivlally
for two values of i - they act differently
V i acts non-trivially
for all values of i - all act the same way
V i acts non-trivially
for all i - two act same, one different
V i acts non-trivially
for all i - all act differently
9 3 27
18 0 0
3 1 3
18 0 0
6 1 6
Proof. Since all representations of a given type are A(G)-
equivalent, they occur with the same multiplicity, so we must
determine this common value for each type. Let the multiplicities
of these types be m0,...,m 6
trivial representation, m I
representation in which V i
i, etc).
It is easy to check that the number of each type of irreduc-
ible appearing in R is as claimed. Thus by counting dimensions
we obtain the equation
(i.e. m 0 is the multiplicity of the
the multiplicity of each irreducible
acts non-trivially for one value of
m 0 + 9m I + 9m 2 + 18m 3 + 3m 3 + 81m 5 + 6m 6 = dim R = 79.
Now consider the action of Pl on R. By 4.1, the dimension
of the subspace of R on which Pl acts trivially is 55. This
subspace is a sum of copies of 32 of the 64 irreducible represen-
tations of G, and it is easy to see that the number of these of
type 0 is I, of type 1 is 7, of type 2 is 5, etc.
50
Proceeding in this fashion for all the subgroups A given in
4.1 yields the linear system
II 9 9 18 3 18 6 1
1 7 5 I0 I 6 2
i 5 3 4 1 2 0
1 6 3 6 0 0 0
1 5 2 5 0 2 1
1 3 3 0 I 0 0~
/ I 4 1 2 0 0 0
i 3 i 2 0 i 0
I 0 0 0 0 0 0
m 0
m 1
m 2
m 3
m 4
m 5
m 6
This (consistent) system has rank 7, and hence a unique solu-
tion, (mo,ml, .... m 6) = (16,3,3,0,I,0,i), yielding the theorem.
From this theorem we may of course determine dim H4(M A) for
any F c A c F(2). There are very many such A (even up to A(G)-
equivalence) so we content ourselves with listing the extreme cases.
Corollary 4"5" Let F c A c F(2) be any subgroup with [A: f] = 2.
Then A is a A(G)-equlvalent to one of the following, and
dim H4(M A) is as stated:
Generator of A/F Dimension of H4(M A)
Pl 55
plP2 51
plq2 45
plP2P3 59
plP2q3 43
plP2qlq3 41
Corollary 4.6. Let F c A c F(2) be any subgroup with
[F(2): A] = 2, so A is the kernel of a homomorphism
~A: F(2)/F --+ {±I}. Then A is a A(G)-equivalent to one of the
following, and dim H4(MA) is as stated:
51
Generators not in Ker(~A) Dimension of H4(M A)
Pl 19
PI' P2 19
PI' q2 16
PI' P2' P3 17
PI' P2' q3 16
PI' P2' ql' q3 17
(Note that here H4(MA) will be a sum of two types of irre-
ducible representations of G, the trivial one and one other. The
six cases of this corollary correspond, in order, to the six non-
trivial types of irreducibles in theorem 4.4.)
We close by considering the question of torsion in the ho-
mology of M A. As we have seen, the only possible torsion is
2-torsion.
Theorem 4.7. Suppose A is untwisted. Then H,(M A)_ _ is torsion
free.
P r o o f . If A is untwisted, then A/r may be written as a product
W I x W 2 x W 3 with W i c V i. (The different W i need not be iso-
morphic.)
From theorem 1.15, we see that M r is rational, and indeed, 0 *
this theorem shows that MF, a Zariski open set in Mr, is iso-
morphic to a Zariski open set in X x X x X = ~ x ~ x ~.
But then M~ is isomorphic to a Zariski open set in
(X/WI) x (X/W2) x (X/W3) , which is itself isomorphic to
x ~ x ~, so M A is rational.
Then by [AM, proposition I], H,(M A) is torsion-free.
52
R e f e r e n c e s
[AM]
[B]
[G]
[LW I ]
[LW2]
[LW 3 ]
IN]
Artin, M. and Mumford, D. Some elementary examples of uni- rational varieties which are not rational, Proc. Lond. Math. Soc. 25 (1972), 75-95.
Bredon, G. Introduction to compact transformation groups, Academic Press, New York, 1972.
van der Geer, G. On the geometry of a Siegel modular three- fold, Math. Ann. 260 (1982), 317-350.
Lee, R. and Weintraub, S. H. Cohomology of a Siegel modular variety of degree two, in Group Actions on Manifolds, R. Schultz, ed., Amer. Math. Soc., Providence, RI, 1985, 433-488.
Cohomology of Sp4(Z) and related groups and spaces, Topology 24 (1985), 291-310.
Moduli spaces of Riemann surfaces of genus two with level structures, to appear in Trans. Amer. Math. Soc.
Mumford, D. Stability of projective varieties, L'Enseignement Math. 23 (1977), 39-110.
Yale University Louisiana State University and Universit~t GSttingen
THE RO(G)-GRADED EQUIVARIANT ORDINARY COHOMOLOGY OF COMPLEX PROJECTIVE SPACES WITH LINEAR 2[/p ACTIONS
L. Gaunce Lewis, Jr.
INTRODUCTION. If X is a CW complex with cells only in even dimensions and R is a ring, then, by an elementary result in cellular cohomology theory, the ordinary eohomology H*(X;R) of X with R coefficients is a free, 7/-graded R-module. Since this result is quite useful in the study of well-behaved complex manifolds like projective spaces or Grassmannians, it would be nice to be able to generalize it to equivariant ordinary eohomology. The result does generalize in the following sense. Let G be a finite group, X be a G-CW complex (in the sense of [MAT, LMSM]), and R be a ring-valued eontravariant coefficient system JILL]. Then the G-equivariant ordinary Bredon cohomology H*(X; R) of X with R coefficients may be regarded as a coefficient system. If the cells of X are all even dimensional, then H*(X;R) is a free module over R in the sense appropriate to coefficient systems. Unfortunately, this theorem does not apply to complex projective spaces or complex Grassmannians with any reasonable nontrivial G-action because these spaces do not have the right kind of G-CW structure. In fact, if G is ~/p, for any prime p, and r / is a nontrivial irreducible complex G-representation, then the theorem does not apply to S ~, the one- point compactification of r 1. Moreover, the 2~-graded Bredon cohomology of S n with coefficients in the Burnside ring coefficient system is quite obviously not free over the coefficient system.
The purpose of this paper is to provide an equivariant generalization of the "freeness" theorem which does apply to an interesting class of G-spaces and to use this result to describe the equivariant ordinary cohomotogy of complex projective spaces with linear :Y/p actions. These results are obtained by regarding equivariant ordinary cohomology as a Mackey functor-vatued theory graded on the real representation ring RO(G) of G [LMM, LMSM]. To obtain such a theory, we take the Burnside ring Mackey funetor as our coefficient ring. Instead of using cells of the form G/H x e n, where H runs over the subgroups of G, we use the unit disks of real G-representations as cells. Our main theorem, Theorem 2.6, then has roughly the following form.
THEOREM. Let G be 2[/p and let X be a G-CW complex constructed from the unit disks of real G-representations. If these disks are all even dimensional and are attached in the proper order, then the equivariant ordinary cohomology H~X of X is a free RO(G)-graded module over the equivariant ordinary eohomology of a point.
To show that this theorem is not without applications, we prove in Theorem 3.1 that if V is a complex G-representation and P(V) is the associated complex projective space with the induced linear G-action, then P(V) has the required type of cell structure. Theorems 4.3 and 4.9, which describe the ring structure of H~P(V), follow from the freeness of H~P(V) . As a sample of these results, assume that p = 2 and V
54
is a complex G-representation consisting of countably many copies of both the (complex) one-dimensional sign representation ,~ and the one dimensional trivial representation 1. Then P(V) is the classifying space for G-equivariant complex line
*p bundles. As all RO(G)-graded ring, i t G (V) is generated by an element c in dimension 1t and an element C ill dimension 1 + A. The second generator is a polynomial generator; the first, satisfies the single relation
c 2 = e2c + ~C,
where e and ~ are elements in the cohomology of a point. If, instead, V contains an equal, but finite, number of copies of A and 1, then the only change in HOP(V ) is that the polynomial generator C is truncated in tile appropriate dimension. If the number of copies of 1 in V is different from the number of copies of A in V, or if p is odd, then the ring structure of H~P(V) is more complex.
Equivariant ordinary Bredon cohomology with Burnside ring coefficients is just the part of RO(G)-graded equivariant ordinary cohomology with Burnside ring coefficients that is indexed on the trivial representations. All of the generators of HOP(V ) occur in dimensions corresponding to nontrivial representations of G. This behavior of the generators offers a partial explanation of the difficulties encountered in trying to compute Bredon cohomology. All that can been seen of HOP(V ) with Z-graded Bredon eohomology is some junk connected to the RO(G)-graded cohomology of a point whose presence in H~P(V) is forced by the unseen generators in the nontrivial dimensions.
Using t t~P(V), tt is possible to give an alternative proof of the homotopy rigidity of linear 2~/p actions on complex projective spaces [LIU]. Moreover, the "freeness" theorem should apply to complex Grassmannians with linear Z /p actions, and it should be possible to compute the ring structure of the equivariant ordinary cohomology of these spaces. Of course, it would be nice to extend the main theorem to groups other than Z/p. Unfortunately, the obvious generalization of this theorem fails for groups other than 7//p. The counterexamples have some interesting connections with the equivariant Hurewicz theorem [LE1]. All of these topics are being investigated.
All of the results in this paper depend on the observation that equivariant cohomology theories are Mackey functor-valued. Therefore, the first section of this paper contains a discussion of Mackey Mnctors for the group 7//p. In the second section, we discuss the RO(G)-graded cohomology of a point, precisely define what we mean by a G-CW complex, and prove our "freeness" theorem. The G-cell structure of complex projective spaces with linear 2~/p actions is discussed in section 3. There the cohomology of these spaces is shown to be free over the cohomology of a point. Section 4 is devoted to the multiplicative structure of the eohomology of a point. The multiplicative structure of the cohomology of complex projective spaces is discussed in section 5. The results stated in this section are proved in section 6. The results on the cohomology of a point stated in sections 2 and 4 are proved in the appendix.
A few comments on notational conventions are necessary. Hereafter, all homology and cohomology is reduced. If X is a G-space and we wish to work with
55
the unreduced cohomolgy of X, then we take the reduced cohomology of X +, the disjoint union of X and a G-trivial basepoint. In particular, instead of speaking of the cohomology of a point, hereafter we speak of the cohomology of S °, which always has trivial G action. If V is a G-representation, then SV and DV are the unit sphere and unit disk of V with respect to some G-invariant norm. The one-point compactification of V is denoted S V and the point at infinity is taken as the basepoint. If X is a based G-space, then NVx denotes tile smash product of X and S V. Unless otherwise noted, all spaces, maps, homotopies, etc., are G-spaces, G-maps, and G-homotopies, etc. We will shift back and forth between real and complex G-representations; in general, real representations will be used for grading our cohomology groups and complex representations will be used in discussions of the structure of projective spaces. If the virtual representation c~ is represented by the difference V - W of representations V and W, then lal = dim V - dim W is the real virtual dimension of a and a G = V G - W G is the fixed virtual representation associated to a. The trivial virtual representation of real dimension n is denoted by n. Recall that the set of irreducible complex representations of G forms a group under tensor product. If 7/is an irreducible complex representation, then r1-1 denotes the inverse of r] in this group. The tensor product of r / and any representation V is denoted 77 V. Many of our formulas contain terms of the form A/p, where A is some integer-valued espression. The claim that A is divisible by p is implicitly included in the use of such a term.
I would like to thank Tammo tom Dieck, Sonderforschungsbereich 170, and the Mathematisches Institut at GSttingen for their hospitality during the initial stages of this work. I would especially like to thank Tammo tom Dieck for suggesting the problem which led to this paper and for invaluable comments, especially on the main theorem, Theorem 2.6.
Equivariant cohomology theories graded on RO(G) are not universally familiar objects, so a few remarks about what this paper assumes of its readers seem appropriate. Equivariant ordinary cohomology with Burnside ring coefficients assigns to each virtual representation c~ in RO(G) a contravariant functor I t~ from the homotopy category of based G-spaces to the category of Mackey functors. It also assigns a suspension natural isomorphism
t t~+v( ,~Vx) =~ tiGX~( )
to each pair (a ,V) consisting of a virtual representation o~ and an actual representation V. The isomorphisms associated to the three pairs (c~, V), (c~, W), and ((~,V + W) are required to satisfy a coherence condition. The functors H~ are required to be exact in the sense that they convert cofibre sequences into long exact sequences. The dimension axiom requires that H~S ° be the Burnside ring Mackey functor and that n 0 tIGS be zero if n E 7/ and n@0. If a is a nontrivial virtual representation, then ~ 0 IIGS need not be zero, but it is uniquely determined by the axioms. Note that because • 0 ttGS is nonzero in dimensions other than zero, the assertion that the cohomology of certain spaces is free over the cohomology of S O is very different from the assertion that the cohomology is free over the coefficient ring. Our cohomology theory is ring calued; that is, any pair of elements drawn from tI~X
56
n~+ZX and t t~X have a cup product wtfich is in a, o . We will also work with
RO(G)-graded, Mackey functor-valued, reduced equivariant ordinary homology with Burnside ring coefficients. This homology theory satisfies the obvious analogs of the cohomology axioms. Also, it has a Hurewicz map, which we use to convert various space level maps into homology classes. Finally, we assume that S O and the free orbit G + satisfy equivariant Spanier-Whitehead duality [WIR, LMSM]; that is, for any e~ in RO(G) there are isomorphisms
I-I~S ° ~H-6~S ° and It~G~ + --~ tI-C~G +.
The proofs of all our results flow from these basic assumptions. In fact, most of the proofs are simple long exact sequence arguments which would be left to the reader in a paper dealing with a g-graded, abelian group-valued, nonequivariant cohomology. One of the points of this paper is that these simple techniques work perfectly well in RO(G)-graded, Mackey functor-valued, equivariant cohomology theories and yield useful results. The one serious demand made of the reader is a willingness to work with Mackey functors. When the group is g /p , these are really very simple objects. Section one is intended as a tutorial on them.
1. MACKEY FUNCTORS FOR Z/p. Since the language of Mackey functors pervades this paper, this section contains a brief introduction to Mackey functors for the groups 7//p. For any finite group G, a G-Mackey functor M is a contravariant additive functor from the Burnside category B(G) of G to the category Ab of abelian groups [DRE, LE2, LIN]. However, since we are only concerned with G = g /p , rather than describing B(G) in detail, we simply note that a g/p-Mackey functor M is determined by two abelian groups, M(G/G) and M(G/e); two maps, a restriction map
and a transfer map
p : M(G/G) -+ M(G/e)
r : M ( G / e ) + M(G/G);
and an action of G on M(G/e). The trace of this action and the composite p r are required to be equal by the definition of the composition of maps in B(G); that is, if x 6 M(G/e) , then
p (x) = gx. geG
The abelian groups M(G/G) and M(G/e) are the values of the Mackey functor M at the trivial orbit and the free orbit; or, if one prefers to think in terms of subgroups instead of orbits, the values of M at the group and at the trivial subgroup. For convenience, we abbreviate G / G to 1 and write M(e) for M(G/e). Frequently the G-action on M(e) is trivial; in these cases the composite pr is just multiplication by p.
A map f : M + N between Mackey functors consists of l lomomorphisn~s
f(1): M(1) ~ N(1) and f(e) : M(e) -+ N(e)
57
which commute with p and r in the obvious sense. The map f(e) must also be G-equivariant. The category ~fft of Mackey functors is a complete and cocomplete abelian category. The limit or colimit of a diagram in ~ is formed by taking the limit or colimit of the corresponding two diagrams consisting of the abelian groups associated to G/G and to G/e. The maps p and r and the group action on the limit or colimit are the obvious induced maps and action.
We wilt describe Mackey functors diagramatically in the form
M(1)
l M(e)
t¢ 0
where M(1) and M(e) will be replaced by the appropriate abelian groups, p and r may be replaced by explicit descriptions of the restriction and transfer maps, and 0 may be replaced by an explicit description of the group action. If p or r is replaced by a number (usually 0, 1, or p), then the map is just multiplication by that number. If 0 is omitted or replaced by 1, then the group action on M(e) is trivial. If p = 2 and 0 is replaced by -1, then the generator of G = Z/2 acts by multiplication by -1.
EXAMPLES 1.1 The following Mackey functors and maps appear repeatedly in our cohomology computations.
(a) The Burnside ring Mackey functor A is given by
Z®Z
(1,P) l l(o,1)
2[
where the notation (1,p) means that the restriction map p is the identity on the first component and multiplication by p on the second. Similarly, (0,1) means that the transfer map is the inclusion into the second factor. For any Mackey functor M, there is a one-to-one correspondence between maps f : A ~ M and elements of M(1). The correspondence relates the map f to the element f(1)((1,0)) of M(1). It follows from this correspondence that A is a projective Mackey functor.
(b) The d-twisted Burnside ring Mackey functor Aid] is given by
ZOZ
(d,P) 1 7 ( 0,1 )
58
where d E g. Note tha t A = A[1]. If d _= + d ' mod p, then there is an i somorph i sm f : Aid] ~ A[d'] of Mackey functors. The m a p f(e) is the ident i ty and if d ' = + d + np, then
f(1)(1,0) = ( + l , n ) E igGT/
f (1)(0 , i ) = (0,1).
I f d = 0 m o d p, then Aid] decomposes as the sum of two other Mackey functors; thus A[d] is only of interest when d ~k 0 rood p. In this ease, it is a project ive Mackey functor . An a l te rna t ive ~-basis for A[d](1) will be used in some of our cohomology calculat ions. To dist inguish the two bases, we denote (1,0) and (0,1) in the present basis by # and r respectively. Select integers a and b such tha t ad + bp = 1. The a l te rna t ive g-basis consists of ~ = a# + b r and ~ = p# - d r . Note t ha t p(~r) = 1, p(~) = 0, and r (1 ) = r . In fact, ~ generates the kernel of p, and r generates the image of the m a p r for which it is named. Of course, c~ depends on the choice of a and b; in our applicat ions, these choices will a lways be specified.
(c) If C is any abel ian group, then we use (C) to denote the Mackey functor described by the d i ag ram
C
°I l° 0
(d) If d 1 and d2 are integers pr ime to p, then there is an i somorph ism
g12: A[dm] ® (77} - , A[d2] G (g}.
Let #i and r i be the s tandard generators for A[di], and let z 1 and z 2 be genera tors of (77)(1) in the domain and range of g~2. Select integers a i and b i such t ha t aid i q- bip = 1, for i = 1 or 2. The m a p g12(e) : 77 -~ 77 is the ident i ty map , and the m a p g12(1) is given by
g l ; ( 1 ) ( t q ) = d 1(a2/*2 + b2r2) + (bl + b2 - blb2P )z2
g12(1)(r l ) = r 2
and
g12(1)(zl) = P#2 - d2r~ - a ld~z>
T h e inverse of g12 is jus t g~l- The existence of this i somorph i sm will explain an appa ren t inconsistency in our descript ion of the equivar iant cohomology of project ive spaces.
(e) Associated to an abel ian group B with a G-act ion, we have the Mackey functors L(B) and R(B) given by
59
L(B) R(B) B /G B G
I B B t2 0 0
Here, ~ : B c -* B is the inclusion of the fixed point subgroup and 7r : B -* B / G is the projection onto the orbit group. The two maps tr are variants of the trace map. The
map t r : B - * B c takes x E B to ~ g x E B e. I f x E B and Ix] is the associated
equivalence class in B/G, then tr : ~ f G --* B is given by
tr([x]) = 2 g x E B. gcG
These two constructions give functors from the category of 7/[G]-modules to the category of Mackey functors. These functors are the left and right adjoints to the obvious forgetful functor from the category of Mackey functors to the category of 2r[G]-modules. We will encounter these functors most often when B is 7/ with the trivial action or, if p = 2, with the sign action. Denote the resulting Mackey functors by L, R, L_, and R_. These functors are described by the diagrams
L R Z
7/ 7/
1 1
L_ P~_ z/2
-1
0
Z U" -1
If C is any abelian group, there is an obvious permutation action of G on C p, the direct sum of p co~es of C. Unless otherwise indicated, this action is assumed
I1 ~ P when we refer to L(C ) or R(C ). These two functors are isomorphic and are described by the diagram
60
C
C p
d 0
where A is the diagonal map. V is the folding map, and 0 is the permutation action.
(f) If M is a Mackey functor, then L(M(e) p) ~ R(M(e) p) is denoted M G.
There are two reasonable choices of a G action on M(e) p, the permutation action or
the composite of the permutation action and the given action of G on each factor M(e). These actions yield isomorphic g[G]-modules, so the choice is not important. The simple permutation action is always assumed here. The assignment of M G to M is a special case of an important construction in induction theory [DRE, LE2] that assigns a Mackey functor M b to each object b of B(G) and each Mackey functor M.
The restriction map p : M ( 1 ) - + M ( e ) ~ M G ( t ) and the diagonal map
A: M(e)-4 M(e )P~MG(e ) form a natural transformation p from M to M G.
Similarly, r : MG(1) ~ M ( e ) + M(1) and the folding map V: MG(e ) ~ M ( e ) p + M(e)
form a natural transformation r : M G -+ M. The Mackey functor A c = L(77 p) is characterized by the fact that, for any Mackey functor M, there is a one-to-one correspondence between maps f : Ao -+ M and elements of M(e). This correspondence relates the map f to the element f(e)((1,0,0 . . . . . 0)) of M(e). It follows that A G is a projective Mackey functor.
G (g) If Y is a G-space, M is a Mackey functor, c, 6 RO(G), and H~(Y;M) and Ha(Y; M) denote the abelian group-valued equivariant ordinary cohomology and homology of Y with coefficients M in dimension a, then the Mackey functor valued cohomology H~(Y; M) and homology H~(Y; M) are described by the diagrams
M) Hg(Y; r,1)
Ha(G × Y; M) H~(G x Y; M)
6
where the maps rr*and rr. are induced by the projection r r : G x Y-+ Y, and the maps rr t and rr! are the transfer maps arising from regarding the projection rr as a covering space. The group H~(G x Y;M) is isomorphic to the nonequivariant cohomology group HIm(Y;M(e)). If r~ is represented by the difference V - W of representations V and W, then, under this isomorphism, the action of an element g of
61
G on H~(G x Y;M) may be described as the composite of multiplication by the degrees of the maps g : S V ~ S V and g : S W - * S W and the actions of g on
HIm(Y; M(e)) induced by the action of g on M(e) and the action of g-1 on Y. Similar
remarks apply in homology. If no coefficient Mackey functor M is indicated in equivariant cohomology or homology, then Burnside ring coefficients are intended.
(h) For any Mackey functor M and any abelian group B, the Mackey functor M ® B has value M(G/H) ® B for the orbit G / H and the obvious restriction, transfer, and action by G. If M* is an RO(G)-graded G-Mackey functor and B* is a Z-graded abelian group, then M* ® B* is the RO(G)-graded G-Mackey functor defined by
(M*® B*) c~ = ~ M z ® B '~. /~+n=a
If a CW complex Y with cells only in even dimensions is regarded as a G-space by assigning it the trivial G-action, then there is an isomorphism of RO(G)-graded Mackey functors
* ~ ® H * ( Y ; 77) H e Y = ~H* S o
which preserves cup products.
For any finite group G, there is a box product operation [] on the category ~Jl of G-Mackey functors which behaves like the tensor product on the category of abelian groups. In particular, ~0l is a symmetric monoidal closed category under the box product. The Burnside ring Mackey functor A is the unit for •. If G = 27/p, then the box product M [] N of Mackey functors M and N is described by the diagram
I-M(1) ® N(1) ® M(e) ® N ( e ) ] / ~
M(e) ® N(e)
0
The equivalence relation ~ is given by
x ® r y ~ p x ® y
r v ® w ,,~ v ® p w
for x C M(1) and y e N(e)
for v C M(e)and w C N(1).
The action 0 of G on M(e) @ N(e) is just the tensor product of the actions of G on M(e) and N(e). The map r is derived from the inclusion of M(e)® N(e) as a summand of the direct sum used to define M rlN(1). The map p is induced by p ® p on the first summand and the trace map of the action 0 on the second.
EXAMPLES 1.2(a) For any integers d I and d2, there is an isomorphism
A[dl]DA[d2] ~-- A[dld2]
62
of Mackey functors.
(b) isomorphism
(c) diagram
If B is a 7][G]-module and M is any Mackey functor, then there is an
L(B)IqM ~ L(B®M(e)).
For any Mackey functor M, the product ROM is described by the
M(1)/(p - rp)
pt ~ 7 r! M(e)
0
where M(1)/(p - rp) is the cokernel of the difference between the multiplication by p map and the composite rp. The maps pt and r t are induced by the restriction and transfer maps for M. In particular, if M = R(B) for some 7[G]-module B, then R[]R(B) ~ l%(B). Also, for any abelian group C, R • < C > ~ < C / p C > .
(d) If p = 2, then for any Mackey functor M, the product R_DM is described by the diagram
M(e)/(image p)
1 - u ~ ) r r
Ct -0
Here ~r: M(e) -~ M(e)/(image p) is the projection onto the cokernel of the restriction map and v: M(e) --* M(e) describes the action of the nontrivial element of G on M(e). The action -0 is the composite of the given action 0 of G on M(e) and the sign action of GonM(e) . In particular, R_[]R ~ L.
(e) For any abelian group C and any Mackey functor M,
< :C:>•M ~ <C®(M(1)/ image r )> .
A Mackey functor ring (or Green functor [DRE, LE2]) is a Mackey functor S together with a multiplication map # : S 13 S --* S and a unit map r/: A ~ S making the appropriate diagrams commute. A module over S is just a Mackey functor M together with an action map ~:SDM--* M making the appropriate diagrams commute. Since the Burnside ring Mackey functor A is the unit for [], it is a Mackey functor ring whose multiplication is the isomorphism A []A ~ A and whose unit is
63
the identity map A --* A. Every Mackey functor is a module over A with action map the isomorphism A [ ] M -~ M. Note that if S is a Mackey functor ring and R is a ring, then the Mackey functor S ® R of Examples 1.1(h) is a Mackey functor ring. Similar remarks apply in the graded case. The cohomology of any G-space Y with coefficients a Mackey functor ring S is an RO(G)-graded Mackey functor ring whose multiplication is given by maps
'~ Y" • t t ~ ( Y ; S ) --* .o.~ t ; ,, I I G ( , S ) • ~+Z/y S~
for ~ and /3 in RO(G).
The following result characterizes maps out of box products and allows us to describe a Mackey functor ring S in terms of S(1) and S(e). This is the approach to Mackey functor rings used in our discussion of the ring structure of the cohomology of complex projective spaces.
P R O P O S I T I O N 1.3 For any Mackey functors M, N and P, there is a one-to-one correspondence between maps h : M [-1N --* P and pairs H = (H1, He) of maps
n 1 : M(1) ® N(1) -* P(1)
H e : M(e) ® N(e) --* P(e)
such that, for x E M(1), y E N(1), z E M(e), and w E N(e),
The second and third of these equations are called the Frobenius relations.
PROOF. The maps H e and h are related by H e = h(e). Given h, H 1 is derived in an obvious way from h(1) using the definition of MV1N. Given H 1 and He, h(1) is constructed from the maps H I and T H e on the two summands used to define M E N ( l ) .
It follows easily from the proposition that, if S is a Mackey functor ring, then S(1) and S(e) are rings, p: S(1) ~ S(e) is a ring homomorphism, and r : S(e) --* S(1) is an S(1)-module map when S(e) is considered an S(1)-module via p. Moreover, if M is a Mackey functor module over S, then M(1) is an S(1)-module and M(e) is an S(e)- module. If we regard M(e) as an S(1)-module via p:S(1)--* S(e), then the maps p: M(1) --* M(e) and T: M(e) --* M(e) are S(1)-module maps.
2. H* ~0 AND SPACES W I T H FREE COHOMOLOGY. Here, we recall Stong's G ° unpublished description of the additive structure of the RO(G)-graded equivariant ordinary cohomology of S °. We use this to show that if X is a generalized G-cell complex constructed from suitable even-dimensional cells, then H~X and H G x are
* 0 free over t t G S . The additive structure of the cohomology HOG + of the free orbit is also described. This is used to show that FI~X and tt.GX are projective over H~S °
64
when X is constructed from a slightly more general class of even-dimensional cells.
* 0 Since 72/2 has only one nontr ivial irreducible representat ion, I t e S is very easy to describe when G = 7//2.
T H E O R E M 2.1. If G = 22/2 and s E RO(G), then
c¢ 0 H e S =
r A, if lal = Is el = o, R, if Isl = 0, [ s e l < 0, and let el is even, R_, if tetl = 0, t a e i _< 1, and t a e i i s o d d , L, if letl = 0, Is el > o, and let Gi is even, L_, if IetI = 0, let et > 1, and let el is odd,
(~), if Isl # 0 arid let eI = 0, (72/2), if letl > 0, I s e l < 0, and ietel is even, (7//2), if le t l < 0, Io, e l > 1, and l e t e l is odd,
~. 0, otherwise.
• o FIGS for various a on The most effective way to visualize tIGS is to display a 0 a coordinate plane in which the horizontal and vertical coordinates specify lete[ and lad respectively. In such a plot, given as Table 2.2 below, the zero values of HeS* 0 are indicated by blanks. The only values in this plot with odd horizontal coordinate are the R_ and L_ on the horizontal axis and the (7//2} in the fourth quadrant .
Even though the representat ion ring of G is much more complicated when p :/= 2, I-I~S ° is completely determined by the integers a and ]o~GI except in the special case where Isl = c~ c ---- 0. In this special case, II~S ° is Aid] for some integer
65
d which depends on a. Unfortunately, because of the isomorphism described in Examples 1.1(b), d is only determined up to a multiple of p. The major source of unpleasantness in the description of the multiplicative structure of the equivariant cohomology of a point and of complex projective spaces is this lack of a canonical choice for d. To explain the relation between a and d, we introduce several relatives of the representation ring. Let R(G) be the complex representation ring of G and RSO(G) be the ring of SO-isomorphism classes of SO-representations of G. Since any real representation of G is also an SO-representation, the difference between RO(G) and RSO(G) is that, in RSO(G), equivalences between representations are required to preserve underlying nonequivariant orientations on the representation spaces. The difference between R(G) and RSO(G) is that elements of RSO(G) may contain an odd number of copies of the trivial one-dimensional real representation of O. Let R0(G ), RO0(G ), and RSO0(G ) denote the subrings of R(G), gO(G), and RSO(G) containing those virtual representations a with Ic~l = [aGt = O. Note that R0(G ) = RSO0(G ). Let R0(G) be the free abelian monoid generated by the formal differences C-r] of complex isomorphism classes of nontrivial irreducible complex representations. Note that R0(G ) is the quotient of R0(G) obtained by allowing the obvious cancellations and that RO0(G ) is the quotient of R0(G ) obtained by identifying conjugate representations. Let A be the irreducible complex representation which sends the standard generator of 7//p to e 2'~i/p. The monoid R0(G) is generated by elements of the form Am _ An where 1 < m , n _< p - 1 . Define a homomorphism from R0(G) to 77, regarded as a monoid under multiplication, by sending the generator A m - A n to m(n-1), where n -1 denotes the unique integer such that 1 _< n -1 _< p - 1 and n(n - 1 ) - 1 mod p. Define functions from RSO0(G ) and RO0(G ) into 77 by composing this homomorphism with sections of the projections from R0(G ) to RSOo(G ) or RO0(G ). Let d~ denote the integer assigned to the virtual representation oe by either map. The sections can not be chosen to be homomorphisms, so the assignment of dc~ to a will not be a homomorphism from RSO0(G ) or RO0(G ) to the multiplicative monoid g. However, the assignment of da to o~ does give a homomorphism from R0(G ) to the group of units (77/p)* of g / p and a homomorphism from RO0(G ) to the quotient (77/p)*/{+1} of (g/p)*. For later convenience, we select our sections so that d o is 1.
Stong's description of the additive structure of * 0 HGS can now be translated into the Mackey functor language of section one.
THEOREM 2.3. If p is odd, then
A[d~J R L
0 JaG s = (7/)
<77/p) (X/p)
0
if if if if if if lal < 0, otherwise
lai = laGt = 0 Ic~l = 0 and taG] <: 0 lal = 0 and la G ] > 0 tal :fi 0 and la c] = 0 tal > 0, lae] < 0, and la Gl is an even integer
laGI > 1, and let GI is an odd integer
66
As in the case p = 2, H~S ° is best visualized by plotting it on a coordinate plane whose horizontal and vertical axes specify faGl and lal respectively, In this plot, given as Table 2.4 below, the zero values • 0 of ItGS are indicated by blanks. The vertical and horizontal coordinates of all the nonzero values, except the (2r/p) values in the fourth quadrant, are even, Notice in the plots for both the odd primes and 2 that the vanishing of * 0 2) is ttGS on the vertical line laGl = 1 (for I~l ¢ 0 if p = unlike its behavior on the vertical lines corresponding to the other odd positive values for lc*c[. These unusual zeroes for H~S ° are the key to our freeness and projectivity results. When G = ?7/pn for n > 1, the corresponding values are not zero, so our techniques do not extend to these groups.
Hereafter, we will often describe elements in H~S ° by their position in these plots• For example, we may refer to the torsion in the fourth quadrant or the copies of (7]} on the positive vertical axis.
(Z/p) (Z/p) (Z/p) (~)
<~/p> (~/p> <~/p} (~>
... (~/p} (~/P) (~/P) (~)
R R R A[d~] L L L
(iV/p} (~/p}
9z) (~/P} (~/P}
(Z/p> {~/p)
TABLE 2.4. H~S ° for p odd.
Recall, from Examples 1.1(f), the new Mackey functor M G which can be derived from any Mackey functor M, and the observation that A 6 = L(~ 'p) = R(gP).
. + r ~ ~S0~ and from this, to compute ttGG . It is easy to check that I-I~G + is ~G~ JG,
67
C O R O L L A R Y 2.5. For any prime p,
• + f A o if lal : 0 HGG =
0 otherwise
, + Proposition 4.12 tells us that IIGG is an RO(G)-graded projective module
* 0 over IIGS , and that. any map
f: * + M* HGG -*
of RO(G)-graded modules over H~S ° is completely determined by the image of (1,0,0 . . . . ,0) E gP = H~(G+)(e) in M°(e).
A generalized G-cell complex X is a G-space X together with an increasing sequence of subspaces X,~ of X such that X 0 is a single orbit, X = tO Xn, X has the colimit (or weak) topology from the X,~, and Xn+l is formed from X,~ by attaching G-cells. We will allow two types of G-cells. If V is a G-representation and DV and SV are the unit disk and sphere of V, then the first type of allowed cell is a copy of DV attached to X , by a G-map from SV to X,~. The second type of cell is a copy of G x e "~, where e rr̀ is the unit m-disk with trivial G action, at tached to Xn by a G-map from G x S m-1 to Xn. For each n, we let J , + l denote the set of cells added to X , to form X,+ 1. Regard a cell DV of the first type as even-dimensional if IV] and ]V G] are even. Regard a cell G x e m as even dimensional if m is even.
T H E O R E M 2.6. Let X be a generalized G-cell complex with only even-dimensional cells.
(a) Assume that X 0 = • and all the cells of X are of the first type; that is, disks DV of G-representations V. Assume also that IV c] >_ IwGI whenever DV 6 J,~, DW 6 ak, 1 < k < n, and IVI > Iwl. Then * + t t c X is a free RO(G)-graded module
• 0 over H~S with one generator in dimension 0 and one generator in dimension V for
each DV 6 J, , , n > 1. The homology I-I,~X + of X is also a free RO(G)-graded • 0 ; " module over H o S vlth generators in the same dimensions.
(b) If X contains cells of both types and all the cells of X of the first type satisfy the condition in part (a), then * + H c X is a projective RO(G)-graded module
l- l* X + l-l* X + • 0 over H~S °. Moreover, ~ G is the sum of one copy of ~ G 0, which is I-ItS or u. G + . o ~ G , in dimension 0, one copy of HGS in dimension V for each DV 6 Jn , and one
. + copy of HGG in dimension 2k for each G x e 2k E J , , n > I. The homology I-I,GX + of X is also a projective RO(G)-graded module over H~S -d and decomposes into the s a n l e summands.
PROOF. Abusing notation, we let J,~+l denote both the set of cells to be added to X~ and the space consisting of the disjoint union of those cells. Let OqJn+ 1 denote the space consisting of the disjoint union of the boundaries of the cells in J,~+l- Associated to the cofibre sequence
The space J,~+i/cgJ~+l is a wedge of one copy of S V for each DV E J~+l and one
• j copy of G+^ S 2k for each G x e 2k E Jn+l. Thus, HG(n+l/O.Jn+l) and
H.Gj( n+i/cgjn+l ) are projective modules over H~S ° with generators in dimensions
corresponding to the cells added to X~ to form X~+ 1. Moreover, if J~+l contains
only type, t ] G ( , J n + l / ( * J n + l ) cells of the first then and HG(j~+l/cgJ,~+i) are free
I:l* X + modules over H~S °. The space X 0 is either a point or tile free orbit G, so ~G 0 and
H.GX0+ are projective, and perhaps free, modules over I t , S ° generated by single
elements in dimension 0.
We will show inductively that the boundary maps 0 in both long exact
sequences are zero. The long exact sequences rnust then break up into short exact * +
sequences which split by the projectivity of HG(Jn+l /0Jn+l) and t IGX, . Thus, by
I I * X + G + , 0 H. X,~ are projective, as appropriate, over HGS , with induction, -~G n and free or
the indicated generators. It follows by the usual colimit argument that HGx + is free,
or projective, with the appropriate generators. Since the map
I-I oe X + l.[ce X + i k G n + l -* .l~/. G n
is always a surjection, the appropriate lim 1 term vanishes, and the cohomology of X,
being the limit of the cohomologies of the X,~, is free (or projective) with the
appropriate generators.
The graded Mackey functors H;(J~+l/cgJ~+l) , I-I.6(J~+i/cgJ~+i) , n* X + S~G 0
G + , + and t I . X 0 are sums of copies of • 0 ttGS and HGG in various dimensions. By
induction, we may assume that . + G + HGX~ and It. X~ are also of this form. To show
that the maps 0 are zero, it therefore suffices to show that they are zero from each
summand of the domain to each summand of the range. For the cohomology
sequence, the four possibilities for the summands and the map between them are:
and
• - 2 k + H G G ~-- H~(G+^S 2k)
• - w 0 ~ t I ~ t t G S = S w
H , - 2 k r _ + ~ H~(G+^ ) G "J : S2k
t t5-Ws ° =~ t i e s w
69
T r * + l z , ~ + c~2rn\ l ~ . + l - 2 r n G + - ~ 116 t, t J A ~ ) ~ " ~ G
T T * + I / , ~ + c~2m\ ~ u * + l - - 2 m ( 2 + - ~ r l G t, kJr A O ) ~ " ~ G "~
* + 1 V ~ I T * + I - V N 0 tt G S = -~G
*+1 V ~ U * + I - V s 0 tIG S = ~-G
lff* X + Here, we use I-I~(G+A S 2k) and H~S w to denote summands of .u. G n isomorphic to
H* G + in dimension 2k or H~S ° in dimension W. The four maps above are all maps G
• 0 of RO(G)-graded modules over I-IGS . Any such map out of * 0 t tGS is determined by . +
the image of 1 E A(1) = H~(S°)(1). By Proposition 4.12, such a map out of IIGG
is determined by the image of (1,0,0 . . . . . 0) E 7/P = H~(G+)(e) . Thus, to show that
T r 2 k + l - - 2 r n / , ~ + x / x the four maps are zero, it suffices to show that the groups zl. G ~o )~,e),
W+l-2m + 1 W+I-V 0 tt G ( G ) ( ) , H ~ + l - V ( s ° ) ( e ) , and tt G (S ) (1 ) are zero. The integers
1 2 k + l - 2 m l and I W + l - 2 m ] are odd and ~ + ttGG vanishes whenever Ictl is odd, so the
first two groups are zero. The integer 12k+l VI is odd and ~ 0 t tG(S )(e) vanishes when
lal is odd, so the third group is zero. For the fourth group, if IVI <IWI, then
tt GW+I-VS0 is zero because ] w G + I - V G] is odd and ] W + I - V ] is positive. Otherwise,
W + l - V o IvGI _> IwGI, and tt G S is zero because Iw + -vq is at most one. An
analogous proof shows that the map (9 in the homology sequence is zero. Note that if
n W + l - V s ° is a result of the Ivl>lwl and IVGI=Iw% then the vanishing of ~G
anomalous zeroes on the I GI = 1 line in the graph of H~S °.
In order to compute the ring structure of the equivariant cohomology of X, we must compare it with more familiar objects, such as the nonequivariant ordinary cohomology of X and X G. If X is a generalized G-cell complex satisfying the conditions of either part of Theorem 2.6, then so is X G. Thus, Examples 1.1(h)
describes H ~ ( X 6 ) + in terms of the nonequivariant cohomology of X G. Since the
tIG(X )(e) is just the nonequivariant ordinary cohomology of X with 7/ group * +
coefficients, the map
p G i* : H~(X+)(1) --* H~(X+)(e) O H ~ ( ( x G ) + ) ( 1 )
70
offers a comparison between H~(X+)(1) and two more easily understood cohomology
rings. This map does not detect the torsion in H~(X+)(1) coming from the fourth
* 0 * G + quadrant torsion in HGS . Moreover, the torsion in tiG((X ) )(1) makes it hard to
compute the image of p @ i*. These difficulties suggest that we also consider the
image of * + ~ + tIG(X )(1)/ torsion in ( H ~ ( X ) ( e ) @ H ~ ( ( X a ) + ) ( 1 ) ) / t o r s i o n . Since * +
I-I6(X )(e) contains no torsion, in the range we are only collapsing out the torsion in
I4~((X6)4-)(1). The most useful comparison map is produced by also collapsing out
* G + the image of the transfer map r from t Ic ( (X ) )(e). The quotient
t I ; ( (xG)+) (1 ) / ( t o r s ion @ im r )
consists of copies of 2 in various dimensions; there is one ~' in the quotient for each
A[d] or (77) which appears in I-I~((XG)+)(1).
For many spaces, including complex projective spaces with linear actions, the cells can be ordered so that Ivl_> Iwl whenever DVEJ,~ , D W ~ J k, and k < n .
* X + t When the cells can be so ordered, there is no torsion in ItG( )( ) in the dimensions of the generators of * + H6X as a module over H~S °. Therefore, the collapsing we have done causes a minimal loss of information. The following result describes the extent
, X + to which tIG( )(1) is detected by p ® i*.
COROLLARY 2.7. Let X be a generalized G-cell complex satisfying the conditions of either part of Theorem 2.6 and let i: X G ~ X be the inclusion of the fixed point set. Then, for any a E RO(G) with Ic~l even, the map
1 4 - _ ~ p • i* : J G( ja&(x4-)(e)
is a monomorphism. Moreover, for any a E RO(G), the map
p@i*: (H~(X+)(1)) / tors ion ~ ~ X4- HG( )(e) @ (H~((XG)+)(1)) / ( torsion @ im T)
is a nmnomorphism.
PROOF. Since the equivariant cohomology of X is the limit of the cohomologies of the Xn, it suffices to show that the result holds for every X,~. It is easy to check the second part for X 0. Assume the second part for X~, and let x be an element of I-IG(X,~+l)(1)/torsmn vanishing under the map into
X + H a ( , ~ + l ) ( e ) ® (H~((xG+l)+)(1)) / ( tors ion @ im r )
X + 1 induced by p G i*. We must show that x is zero. The group ttG( n+l)( ) is the
71
direct sum of the groups I-I~(Jn+z/0J,+l)(1 ) and ~ + t t6(Xn)(1) , and this decomposition
is respected by the map p ® i*. Thus, x is the sum of classes y and z in
o¢ X + • It~(J,~+z/C0Jn+~)(1)/torsion and t i c ( ,~)(1)/torsmn, respectively, which vanish
under the analogous maps. By our inductive hypothesis, z is zero. Since J,~+l/COJ,~+l is a wedge of copies of S V and G+^S ~k for various V and k, y vanishes by our remark about X 0. Thus, x is zero. The proof of the first part is similar. For this part, we must assume that ic~l is even because the map p O i* does not detect the torsion in the fourth quadrant of H~(S°)(1).
3. THE COHOMOLOGY OF COMPLEX PP~OJECIVE SPACES. As an application of the results from section two, we show that the cohomology of a complex projective
" * S O Let V be a finite or countably infinite space with a linear action is free over .u. G . dimensional complex G-representation and let C* be C - {0}. The complex projective space P(V) with linear G-action associated to V is the quotient G-space ( V - {0})/C*. Note that if W C V, then P(W) may be regarded as a subspace of P(V). If V is infinite dimensional, then we topologize V as the eolimit of its finite dimensional subspaces W; the quotient topology on P(V) is then the same as the colimit topology from the associated subspaces P(W). To describe the cohomology of P(V), we must
write V as the sum ~ ¢i of irreducible complex representations (including possibly i = 0
the trivial complex representation). Of course, if V is infinite dimensional, then n = oo. Points in P(V) will be described by homogeneous coordinates of the form
<x0, xl, x2 . . . . . x~), xi e ¢i
with the conventions that not all of the x i are zero, and if V is infinite dimensional, that all but finitely many of the x i are zero. Each element of the group G acts on each homogeneous coordinate of P(V) by multiplication by a complex number. Therefore, if all the irreducibles in V are isomorphic, then the action of G on P(V) is trivial. Moreover, if r] is any irreducible complex representation, then P(V) and P(r 1V) are isomorphic G-spaces. If 7/ and ¢ are irreducible complex representations, then P(rl) is just a point and P(r I O ¢) is G-homeomorphic to the one-point compactification of either r] -1 ¢ or 7/¢-1.
Since we have selected a eolimit topology on P(V) when V is infinite, to show that P(V) is a generalized G-cell complex for any G-representation V, it suffices to
k - 1 show this when V is finite dimensional. Let V k be the representation ~ ¢i and let
i = 0 W be the representation -1 ¢,, V,~_ 1. Describe points in the unit disk DW by complex
-1 coordinates (x0, xl, ... ,xn_l) , with x i C Cn ¢i" Define a map f: DW -~ P(V) by
f((xo, x 1 . . . . . Xn_l) ) = <Xo, Xz, x 2 . . . . . xn_l, 1 - E txil2)" i = 0
The tensor product with ~ n I is inserted in the definition of W to ensure that the map f is equivariant. The image of SW in P(V) lies in the subspace P(Vn_I) of P(V), and f is a homeomorphism from D W - SW to its image in P(V). Thus P(V) is formed
72
from P(Vn_l) by adjoining the G-cell DW along the map f l S W : SW-* P(V~_~). Working backwards through the sequence of representations Vk, we conclude that P(V) is a generalized G-cell complex with cells the unit disks of the representations
¢~lVk for 1 _< k _< n.
In order to show that the equivariant cohomology of P(V) is free over H~S °, we must show that the cells of P(V) can be attached in an order satisfying the condition in Theorem 2.6(a). This proper ordering of cells is derived from a careful ordering of the set q~ of irreducible summands of V. Since the remainder of our discussion focuses on ~, we write P(~) for P(V). An ordering ¢0, ¢~, ¢2 . . . . of the
elements of • is said to be proper if the number of irreducibles in the se t {¢i}O<_i<_k-1 isomorphic to ¢~ is a nondecreasing function of k. For example, if ¢ and r} are distinct complex irreducibles and q5 consists of two copies of ¢ and one of q, then r], ¢, ¢ and ¢, r], ¢ are proper orderings of ~, but ¢, ¢, 7/ is not. The dimension of
lk--1 the fixed subrepresentation of the representation ¢~- ~ ¢ i is the number of
i=0 irreducibles in the s e t {¢i}o<i<k-1 isomorphic to Ck- Thus, if q~ is properly ordered, then the cell structure described above satisfies the conditions of Theorem 2.6.(a).
PROPOSITION 3.1. If ¢0, ¢2, ¢2, -.. is any ordering of the elements of a set • of irreducible representations, then p(dp) is a generalized G-cell complex with cells the
unit disks of the G-representations ¢ - tk-1 k ~ ¢~, for k > I. Moreover, H~P(~5) + and i = 0 * 0
I I~P(~) + are free RO(G)-graded modules over I t , S . If the ordering of ~ is proper, then the homology and cohomology of P(~) are each generated by one element in
k-1 dimension zero and one in each of the dimensions 6~-1 ~ ¢i , for k _> 1.
i=0
The G-fixed subspace of P(q~) is a disjoint union of complex projective spaces, one for each isomorphisnl class of irreducibles in ~5. The (complex) dimension of the complex projective space in P(e)) a associated to the irreducible ¢ is one less than the multiplicity of ¢ in ~. Thus, the effect of properly ordering the irreducibles is that the maximal dimension of the components of the G-fixed subspace of P({¢i}0</_<k) increases as slowly as possible with increasing k.
REMAl~KS 3.2. Our description of the eohomology of P(~) contains one apparent anomaly. Suppose that (, r/, and 4) are distinct complex irreducible representations and • = {(, q, 6}. If we assign the proper ordering (, r/, ¢ to ~, then we find that the generators of * + HGP((P ) are in dimensions 0, r] -1 ¢, and ¢-1 (¢ • r/). However, if we select the proper ordering 6, (~, q, we find that the generators are in dimensions 0, ¢-1 6, and r] -1 (6 O (). In particular, the cohomology in dimension r]-1¢ must be A ® <2[) ® <2[) if we use the first set of generators, and A[d] ® <N> ® <7/) if we use the
second, where d is the integer associated to the element rl - t ( - ( - 1 ¢ of RO0(G).
There is no contradiction in these two claims about the cohomology in dimension
7 3
r / - l ( because these two Mackey functors are isomorphic by Examples 1.1.(d). The apparent difficulties in all the other dimensions are resolved in exactly the same way.
This example illustrates the latitude that one has in selecting the dimensions of the generators of the cohomology of P(~) for almost any set • of irreducibles. This latitude is necessary because, for most ~, there are many proper orderings and a choice of a proper ordering corresponds to a selection of the dimensions of the generators.
It would be nice to have some simple cohomology invariants of P((I,) which could be used for problems like comparing the cohomology of projective spaces with different G-actions. The fact that the dimensions for the cohomology generators don' t provide such an invariant is a disappointment. However, one invariant related to the dimensions of the generators is readily available. Select a proper ordering of ¢5 and plot the dimensions c~ of the resulting set of generators of I-I~P(~) + on a
coordinate plane whose horizontal and vertical axes indicate laGI and I~1,
respectively. The dimensions lie on a stair-step pattern whose foot is at the origin. This plot is an invariant of P(q@ The height of the steps in the plot decreases, or remains constant, as one goes up the steps (that is, moves in the direction of increasing laGI and loci). The height remains constant only if irreducible types appearing in (I) have equal multiplicity. The step-like structure of the plot reflects a filtration on (I) which plays an important role in our discussion of the ring structure
, + ofttGP((I) ) . An increasing filtration
0 = qS(0), ~(1), (P(2) . . . . . ~(r) . . . .
of the set • is said to be proper if, for every r and every complex irreducible ¢, the number of irreducibles in (I)(r) isomorphic to ¢ is the lesser of r mad the number of irreducibles in • isomorphic to ¢. Any two proper filtrations of ~5 differ only by an interchange of isomorphic irreducible complex representations, so there is essentially only one proper filtration of ~. The steps in the plot of the dimensions of the generators are in a one-to-one correspondence with the stages in the filtration of ~. The height of the step corresponding to filtration level r is the number of elements in ~5(r) - ~(r - 1).
4. CUP PRODUCTS IN • 0 H G S . Here we describe the multiplicative structure of • 0 ttGS . We begin with the case p = 2, which is due to Stong.
DEFINITIONS 4.1. Let ( be the real one-dimensional sign representation of
G = ?7/2. The identity element 1 in A(1) = H~(S°)(1) is the identity element of the
RO(G)-graded Macl(ey functor ring tIGS* 0. Let n E It~(S°)(1) be 2 - rp(1) . Observe
that n ' - = 2n. Let ~ E H~(S°)(1) be the Euler class; that is, the image of
1 E H~(S°)(1) under the map induced by the inclusion S ° C S ¢. Select a
74
nonequivariant identication of S C with S 1 and let h - ( • H~-i(S°)(e) -~tt~(S~)(e)
and L(_ 1 • H~-a(S°) (e)~H~(S~)(e) be the images of p ( 1 ) • H~(S° ) (e )~H~(S~) (e )
under the maps induced by this identification. Let ( • tt~<-2(S°)(1) be the unique
element with p ( ( ) : Q_> The elements 1 and n generate the abelian group
H~(S°)(1) and the Mackey functor H~S °. Each of the elements e ~, (~ , and era( ~, a 0 for m, n _> 1, generates the abelian group ttG(S )(1) and the Mackey functor ~Gn~S° in
the appropriate dimension a. For m > 1, the element L~_ i or L~_ 1 generates the
abelian group H~(S°)(e) in the appropriate dimension and L~_< generates the Mackey
functor . o I-IGS in the appropriate dimension. For m > 2, r(L~_~) generates the
• 0 1 abelian group I-IG(S )( ) in the appropriate dimension.
LEMMA 4.2.
, 2 n + 1 , (2n+l)(l-() 0 r t h - < ) 6 t t G ( S ) ( 1 )
are infinitely divisible by e 6 H~(S°)(1); that is, for m elements
and
The class g • H~(S°)(1) and, tbr n _> 1, the classes
_> 1, there are unique
e - ' ~ 6 tt~'~¢(S°)(1)
- m / 2 n + l ~ . 2 n + l - e T i L l _ ~ ) 6 H G (2n+m+l)¢(sO)(1)" "" "
such that
e m ( e - m ~ ) ~ a n d . . . . . . . 2~+1,, , 2~+1, = e (,e r(q_¢ )) = riq_ ().
Moreover, each of the elements e-m,~ o r e--r*~T(/~ 2 n + 1 ) generates the abelian group H~(S°)(1) and the Maekey functor * 0 tlGS in its dimension.
THEOREM 4.3. The elements
e 6 H~(S°)(1)
q_< ~ ~-((S°)(e)
q-1 e l{~-~(S°)(e) 2 ( - 2 o ettG (S)(1)
~-~ ,~ e t t ~ ' ~ ( s ° ) ( 1 ) ,
and
f o r m _> i,
- - m z 2 n + I \ ~ 2 n + 1 g r t , t l _ ( ) E H G - ( 2 n + m + l ) ( ( S ° ) ( 1 ) , f o r m , n _> 1,
75
• 0 RO(G) -g raded Mackey functor a lgebra over the Burnside generate I tGS as an Mackey functor ring A. The only relat ions a m o n g these elements, other than those forced by the Frobenius relat ions or the vanishing of I-I~S ° in various dimensions, are genera ted by the relat ions
p(~) = o
*1-¢ ~¢-I = p(1)
r(q_<) = 0 / 2 r e + l \ r(~¢_ ~ ) = O,
{0 ~-(,7_~) ~(,?_~) = 2~(,7._~),
2 e ~ = O
p ( c - ~ ) = O,
( ~ - ~ ) ( , - ~ ) = 2~-(~+~>~,
/ 2 r~+lx 2e - m r ( t l _ ~ ) = O,
p(C.-m z 2 n + l ~ \ rk t l _ ¢ )) = O,
- m 2 n + l = ( ~ l - r n / 2 r t + l x
(¢-m , 2 ~ + 1 , , ( - % ) 0, r t t l - ¢ )) =
and
-rr~ 2 n + l (~ ~-('1-~ )) 6 - m / 2 n - l x
T ( L I - ( ) ,
for m _> 0,
f o r m _> 1,
if m or n is odd,
if m and n are even,
f o r m > 0,
for m >__ 1,
for m, n > 0,
for m _> 0 and n > 1,
for m _> 0 and n _> 1,
for m, n _> 1,
f o r m , q >_ 0 a n d n > 1,
f o r m >_ 0 a n d n >_ 2.
R E M A R K S 4.4. (a) The last relation indicates that , for m _ 0 and n _> 1, ~ - m i 2 r t + l x r t , l _ ¢ ~ is infinitely divisible by ~. Thus, we can think of all the e lements in
the four th quad ran t of the graph of * 0 t tG(S ) as being derived f rom r(L3 ¢) via division by powers of e and ~. One mnemon ic for the effect of e and ~ on the e lements in the
four th quad ran t is to denote the nonzero e lement in H~-"~ -~" (¢ -1 ) (S° ) (1 ) , for m _ 2
and n _> 1, by e - m ~ - " w, where w is regarded as a ficti t ious e lement in d imension 1. T h e reason for selecting a fictit ious e lement in dimension 1, instead of the actual e lement in d imension 3 - 34, is discussed in Remarks 4.10(b).
(b) For p = 2, the e lements + ( 1 - r p ( 1 ) ) in A(1) are units, and l - r p ( 1 ) appears in the fo rmula describing the a n t i c o m m u t a t i v i t y of cup products . For any
I-I i + J ( X + I-l-m + n i X + then G-space X, if a E z~ G and b E -~'G ,
76
a b = ( - 1 ) i v * ( 1 - r p ( 1 ) ) J n b a .
The generators L1_¢, t¢_1, e, e - n n , and c - m , 2n+l, r~q_¢ ) are in dimensions where the
behavior of this nontrivial unit matters. Of course, since e - ~ ; 2~,+1, rt~l_ ¢ ) has order 2, any unit acts trivially on it. It is easy to check that
(1-rp(1))~l_¢ = - t1-¢ and ( 1 - r p ( 1 ) ) L ¢ _ 1 = -#¢-1"
This action of 1 - rp(1) on LI_ ¢ aI]d re_ 1 never affects cup products in tI~S ° because it is always balanced by the ( -1) 'm term in the commuta t iv i ty formula. However,
n* S o where the effects of this unit on q_¢ and re_ 1 are there are algebras over -~-G visible. The unit 1 rp(1) acts trivially on e and e - ' ~ . This shows up dramatical ly in ~a* S O The elements e and e->~+l~ are odd-dimensional, so our intuition about
a ,a , G •
graded algebras Dom the nonequivariant context suggests that their squares should vanish, or at least be 2-torsion. In fact, the squares are not torsion elements, an apparent anomaly possible only because the action of 1 - rp(1) is trivial. The overall effect of the actions of the units of A on tile generators of * 0 I-IGS is that t t~S ° is commuta t ive in both the graded and the ungraded seuse.
When p is odd, several complications ill the multiplicative structure of I-I~S ° arise from the greater complexity of RO(G). Tile most obvious are a host of sign problems coming fl'om the identification of representations with their complex conjugates. Initially,. we resolve these sign problems by grading HG S , 0 on RSO(G) instead of RO(G). In Remark 4.11, we explain steps which must be taken to pass back to an RO(G)-grading. The most serious complication arises from the misbehavior of the integers da associated to the virtual representations c, in RSO0(G ). One way to deal with this complication is to avoid it. This can be done very nicely if one is only interested in • 0 HGS . Because of the intuition this approach offers, we outline it as an introduction to the odd primes case.
The stable homotopy groups reds °, for ~ E RSO0(G ), have been studied extensively by tom Dieck and Petrie [tDP], and the stable Hurewicz map
h: ~r~_~S °-~tIG_oS ° ~--u~S° - - .~,,,L G
is an isomorphism [LE1] if 3 C RSO0(G). Thus, many of their results can be applied to homology in the appropriate dimensions. They have shown that the multiplication map
° + ze+ G s o
is an isomorphism for any /3 E RSO0(G ) and any 7 E RSO(G). By similar reasoning, the multiplication map
0 01-1 y 0 H~+~S0 I-t~S ttGS -~
is an isomorphism under the same conditions on /3 and 7. Thus, to understand all of I-I~S °, it suffices to understand the part of * 0 tlGS which tom Dieck and Petrie have already described and the part indexed on some subset of RSO(G) complementary to RSO0(G ). Recall that k is the irreducible complex representation that takes the
77
standard generator of 7]/p to e 2'ri/p. Let RSOz(G) be the additive subgroup of RSO(G) generated by 1 and I . As an additive group, RSO(G) is the internal direct sum of RSO0(G ) and RSOx(G). To complete our description of II~S °, it suffices to describe that part of it indexed on RSOx(G). This part is almost identical to H~S °
* 0 for G = ~/2. Consider the description given above of that part of I l l s for p = 2 indexed on the additive subgroup of RO(i7/2) generated by 1 and 2C. Replace 24 by
f . . . . 0 for p ,k and all the other 2's by p's. The result is a description of the part o r t6~ odd indexed on RSO:~(G). This approach describes * o t t c S as the graded box product of two subrings indexed on complementary subsets of RSO(G). The unpleasant behavior of the integers d~ is buried in the computations of the box products.
Unfortunately, because of peculiarities in the dimensions of the algebra generators of H~P(V) +, this description • 0 of HaS as the box product of two subrings can not be used to describe the ring structure of the cohomology of complex projective spaces. Thus, we offer an alternative description of the ring structure of It* S O for p odd. In section 2, we defined a function from RO0(G ) to Z using a G section of the projection from R0(G) to RO0(G ). Since we are now working with RSO0(G ) instead of RO0(G), we define an analogous function from RSO0(G ) to 7] using a section s: RSO0(G ) -* Ro(G) of the projection from t~0(G ) to RO0(G). We insist that s(0) = 0 and that our original section RO0(G ) -* R0(G) factor through s.
DEFINITIONS 4.5. (a) If ct E RSO0(G ) and s(a) = ~ ¢ i - r h , then we wish to
define an equivariant map #~: S ~ i - * S ~¢i with nonequivariant degree d~. If
a = I m - I '~ with 0 < m,n < p and n -1 is the unique integer such that 1 _<n -1 _<p- 1
and nn -1 ~ 1 rood p, then ~t~ is the extension to one-point compactifications of the
complex power map z -~ z "~('~-1), for z E C. In general, we identify S x t i and S "~ni
with Ai Sci and A S ~i, respectively, and take the smash product of the appropriate
complex power maps to obtain the equivariant map #~ from S ~¢i to S ~ ' i with
nonequivariant degree d~. Also denote by #~ the image of this map in It~(S°)(1)
under the Hurewicz map. Clearly, if the 8i and the ~?i were paired off in a different
order, then a different map from S ::el to S ~"i would be obtained. However, the
maps coining from the two pairings would be equivariantly homotopic and so would
give the same element in H~(S°)(1).
(b) Let a be an element of RSO(G) with lal = 0. Then a must be
represented by a s u m E~i-?]i, where the ¢i and r h are irreducible complex i
representations, some of which may be trivial. Since the 8i and r h are complex
representations, they have canonical nonequivariant orientations. Combine these to
produce a nonequivariant identification t~ of S ~¢i with S ~ni which is unique up to
78
homotopy. Let Le also denote the image of this identification in I-I~(S°)(e). The
resulting cohomology classes L~ are then independent of the ordering of the ¢i and
the Vi. The abelian group H~(S°)(e) is generated by L~. If lc~GI > 0, then 7"(~)
generates the abelian group H~(S°)(1) and L~ generates the Mackay functor ltGS~ 0.
(c) If c~ E RSO0(G), then in HGS ~ 0,
p(#~) = d~ ~ and pr(L~,) = p ~ .
We have already asserted that I t , S ° is A[d~]. Under this identification, #~ and
r ( t~) become the elements /~ and r of A[d~](1) and ~ becomes 1 E 7/ = A[d~](e).
There is a unique integer b ~ s u c h t h a t d _ ~ d ~ + b~p = 1. Let ~ = p # ~ - d ~ T ( L ~ )
and G~ = d - ~ # ~ + b~T(L~). Then, cr~ and ~ form an alternative Z-basis for
tt~(S°)(1).
(d) Let /3 be an element of P~SO(G) with 1/31 > 0 and 19GI = 0 There exist
an c~ in RSO0(G ) and a G-representation V such that V G = 0 and /5 = c~ + V. Let
ep E linG(S°)(1) be the image of #c~ E t t ; (S°) (1) under the map from It~(S°)(1) to
ttZG(S°)(1) induced by the inclusion S o C S v. In Lamina A.11, it is shown that this
Euler class eZ is independent of the choice of the decomposition of/5 into the sum of
the representation V and the element c~ of RSO0(G). The class eZ generates the
abelian group HZG(S°)(1) and the Mackay functor tt~GS °.
(e) If I~l = 0 and < 0, let be the unique element of I-I~(S°)(1) with
p({~) = L~; this class generates the abelian group H~(S°)(1) and the Mackay functor
tt s °.
When p is odd, it is harder to pick a multiplicative basis for the torsion in the fourth quadrant of the graph of u* S o In each dimension there is a choice of " ~ ' G "
p - 1 generators, instead of a single nonzero element. Moreover, since these torsion elements are not tied by an Euler class to elements on the positive horizontal axis, there is no way to base the choice of a generator on choices already made for the axis. The following lemma justifies the procedure we employ to select multiplicative generators for the fourth quadrant.
LEMMA 4.6. Let 13 be an element of RSO0(G ) and let a, 7, and 6 be elements of RSO(G) such that
79
and
lal = I~GI = 0,
I~1, la~l < 0,
Ivl > 0,
Is~l > 3,
Io~GI is odd.
If x is any nonzero element in I-IG(S~ 0)(1), then p z x is a generator in ttG+Z(S°)(1). Moreover, x and #Z x are uniquely divisible by both c-~ and ~e"
DEFINITIONS 4.7. Select a generator in I-I~-~a(S°)(1) and denote it by ~'3-~a- If
~ = 1 - m ( ) , - 2 ) - n l , for m, n_> 1, then let us be the unique element in H~(S°)(1)
such that
For any o~ C RSO(G), there are unique integers m, n, and q such that q = 0 or 1
and
c t - [ q - m ( , k - 2 ) - n & ] E RO0(G).
Denote by < a > the element q - m ( , k - 2 ) - n , k associated to a by these conditions. If
ct E RSO(G) with lal < 0, laGI _> 3, laGI odd, and a :~ < a > , then define
uo" E tt~(S°)(1) by
P'o" : t / o ' _ < a > b ' < o ' > .
The element uo" generates the abelian group H~(S°)(1) and the Mackey functor a 0 ttGS .
LEMMA 4.8. If a E RSO0(G), then /go" E It~(S°)(1) is divisible by ee, for any
fl e RSO(G) with ]fl[ > 0 and ]fiG] = 0; that is, there is a unique element
~71/gs e ~ 9(s°)(1)
such that
~5,8 ((T~ 1 /gO,) = /go',
The element e~ 1/go" generates the abelian group I-I~-P(S°)(1) and the Mackey functor
~; -e s0 .
80
T H E O R E M 4.9. The e lements
and
c~ o ~ e ttG(s )(1), 0 ~ ~ t tG(s )(e),
% • t t~ (S°) (1 )
~ - 2 • t t~-~(S°) (1)
~:_~ e e~-~(s°)(e)
~;~,~o • I~;~(s°)(1),
a 0 ~,~ c t tG(s )(1),
for c~ = : t : ( A n - A ) , with 1 < n < p,
for o~ =: t : ( )~n- ,x) , with t < n < p,
f o r m > 1,
dimensions, are generated by the relat ions
p ( ~ ) = d ~ ,
p(~e) = 0,
T(~) = p¢~ ,
~- ¢~ = ¢~+e,
~ ¢ ~ = d~4~+~,
for a c RSOo(G);
for c~, fl E RSO0(G);
for 191 > 0 and 191 = 0;
for lal, Ifl] > 0 and
I~1 =1~[_=0;
for ~ e a S O o ( 6 ) , f~l > o,
and I~ l - - O;
for lal = 0 and laG[ < 0;
for Iod = 0 and I~I < 0;
for la'I = I~I = 0 and
l~el, b q < 0;
for a E RSO0(G) , t/31 = 0,
and triG] < 0;
• 0 RSO(G) -g raded Mackey functor a lgebra over the Burnside generate H a S as an Maekey functor ring A. All of relat ions a m o n g the e lements of * o I t6S , o ther than those forced by the Frobenius relat ions or the vanishing of • 0 I-I6S in var ious
for a = l - - i n ( A - 2 ) - n A , w i t h i n , n_>1,
p ee {~ = 0,
e/~ {~, = da_ ~ e-r {~,
p ( { ~ ; l ~ a ) = O,
/*3' ( e ; 1 /~oe) = ,£;1 K:°c+7 '
eft (£.;1 /~0,) ~- KOe,
e7 (e~ 1 ~ ) = e~l_~ ~c~,
-1 (e~l~a)(c;l '~6) = PQ~+-r ha+o,
pu~ = O,
p ( ~ ) = o,
81
for ~ = t g q : 0, i ~ q < 0,
~nd I,< > 0;
for Iod = [81--1t3GI = 17GI = 0,
I~q, leq < 0, I~1,1:1 > o, and a + / 3 = 7 + 6 ;
for a, ~5 E RSOo(G),
Ieq=i:q=o, I~t, 171 > 0, and
a + 7 = , ~ + 6 ;
for ~ 6 RSO0(G), I s g = 0 ,
~nd t~1 > 0;
for a, 7 E RSOo(G), I~g ;o, and 191 > o;
['or c~ E RSOo(G ), I f lGl=0,
and I~l > 0;
for a E RSOo(G),
I ~ q - - I ~ q = o , and
]/~1 > 171 > 0;
for a, 6 E RSOo(G),
19[ = I'~g ; o ,
and 1/31, [7I > 0;
for t~1 < 0, ic~GI > 3, and
l~q odd;
f o r l a l < O , [aG[>a, and
I~q odd;
for /3 E RSOo(G), lat < O,
IaGI _> 3, and ioeGI odd;
82
(3 ~'c~ --~ /"c,+~,
( E ; 1 //;7) /JOe = O, for
tc~ I.fi ~ tcx+fi ,
for Io~-4- ~1 < 0, IsGI ~ 3,
lonG] odd, I'~1 > 0, and
I~GI--0;
for ,~, < 0, Io~ ~ + ,~GI > 3,
[czGI odd, I~1 = o, and
19Gi < 0;
7 E RSO0(G), Ic~l < 0,
IsGI >_ a, I~GI odd,
[fiG[= 0, and I~1 > 0;
i~1 = l ~ l = 0 . for
REMARKS 4.10. (a) For p odd, the only units in A(1) are +1. The only generators in odd dimensions are the v~. Since v~vZ is zero for any c~ and fl, no sign problems occur in commuting products in H~S °. Thus, H~S ° is commutative in both the graded and ungraded senses.
(b) As an alternative to using the v~ as a basis in the fourth quadrant, one
may define elements e71~1co in ttG-~-QS°)(1), for l a l = [ / 3 G ] = 0 , l aG]<0 , and
191 > 0, by
( 7 1 ~ 1 C 0 = d~,_<~>//1-a'--fi"
Here, aa is regarded as a fictitious element in dimension 1 which is divisible by any product ~<, e/~. We employ a fictitious element because there is no canonical choice for the dimension of an actual element. The relations satisfied by the elements e~z~lo~ are
- 1 - 1 for ,~, = I~ G] : I 'r~ I : O,
191 > 171 > o,
and [a G] < O;
for I~, = rTJ = f~GJ = o,
I~1 < I,r~ I < 0,
and ]fi[ > O;
83
#7 (e71~7'1w) = d<v>-~ e7 l ~ a !~ w '
for 7 E RSO0(G),
= 1/3 1 = o, l s e i < o,
and 1/31 > 0;
for 7 e RSO0(G),
and 1/31 > 0.
The one difficulty with this alternative basis is that if a + / 3 = 7 + 5, then e~l~gl~o and e}-l~71w are in the same dimension, but they need not be equal. In fact,
(c) Observe that in the formulas for the product of #~ with any of e/~, e~ 1 ~ , or v5 there is no de,, but there is such a constant in the formula for the product p~ f~ . On the other hand, er~ f/~ = ~ + ~ , but there is a d -~ in the formula
for the product of ~ , with any of e/~, e~ 1 ~v, or t@. This difference in the behavior
• 0 1 of the elements #~ and er~ of HG(S )( ) reflects the fact that there is a conjugacy
class of subgroups of G associated to any well chosen element of any G-Mackey functor M for any finite group G. This association is based on the splitting of M which occurs when M is localized away from the order of G. This splitting can not be observed directly before localization, but it can be seen indirectly in the association of subgroups to well chosen elements in the Mackey functor. The elements #~ , e~,
e~ 1 ~y, and v~ are all associated to the subgroup G of G, and products of pairs of
them behave nicely. The elements era and ~Z are associated to the trivial subgroup, and their product is nice. However, the product of elements associated to two different subgroups will either be zero or involve some fudge factor like a d a . We have introduced both #a and era so that, when one of these elements is needed in our description of the relations in H~P(V) +, we can always choose the one that will give us the simpler formula.
• 0 REMARKS 4.11. In order to explain the passage from an RSO(G) grading on HGS
to an RO(G) grading, we must first clarify what is meant by the assertion tha t HGS* 0
is RO(G)-graded. The assertion does not mean that , for a C RO(G), ~ 0 ] t6S can be described without reference to a choice of a representative for a. Rather it means that if V I - W 1 and V 2 - W2 are two representatives for a and I-I 1 and H 2 are the
values of ~ 0 FIGS obtained using these representatives, then we can construct an
isomorphism between I-I 1 and H 2 in a natural way from any isomorphism
f: V2 O W1 ~ V1 G \¥~ of representations illustrating the equivalence of V 1 - W1
and V 2 - W 2 in RO(G). This is exactly what we mean when we say that nonequivariant homology is 7/ graded. To define the nonequivariant homology group H~X, we must pick a standard n-simplex. Different choices of the n-simplex lead to
8 4
different groups, as anyone who has been embarrassed by an orientation mistake knows all too well.
Let fl---- V 2 G W 1 -V~ ® W 2 and let f denote the image of f in Ji~(S°)(1). Then the isomorphism from H 1 to H 2 is just multiplication by f. To provide a means of computing the effect of this isomorphism, we write f in terms of the
s tandard generators of tI~(S°)(1). The map f induces a map 1¢ 3 between the fixed
point subspaces of the representations. If nonequivariant orientations are choose for their domains and ranges, then the maps f and fG have well-defined nonequivariant degrees. It follows from Lemma A.12 that
(deg f ) - (deg fG)d~ = (deg fG)/2~ + p r(L/~).
The structure of • + • 0 HGG as an algebra over JIGS follows easily from our • 0 results on ttGS and the description of the additive structure of * + JIGG given in
section 2.
. 0 • + P R O P O S I T I O N 4.12. As an RO(G)-graded module over ttGS , is HGG generated
by the single element ~ = (1, 0, 0 . . . . . 0) of J I ~ ( G + ) ( e ) = Z p. Moreover, for any
RO(G)-graded module M* over H~S °, there is a one-to-one correspondence between
maps f: * + M* HGG --* of RO(G)-graded modules over H~S ° and elements in M°(e).
This correspondence associates the map f with the element f(e)(~) of M°(e). Thus,
• + * 0 JIGG is a projective RO(G)-graded module over ttGS .
PROOF. Unless lal 0, ~ + ~ + = ItG(G ) = 0 . If l a l = 0 , then t ~ generates HGG as a
module over A. Thus, ~ generates H* G + • 0 ~-G as an RO(G)-graded module over JIGS , , +
and any RO(G)-graded module map f: HGG + M* is determined by f(~). On the
other hand, recall the observation from Examples 1.1(f) that a map from A G to any
Mackey funetor N can be specified by giving the image of (1, 0, 0 . . . . . 0) E AG(e ) in
N(e). Let m be an element of M°(e). For each o E R O ( G ) with I c d = 0 , Lore is in
c~ + M~(e) and there is a unique map f ~ : H G G + M ~ of Maekey functors sending
tc~b C I-I~(G+)(e) to t a m C M~(e). These maps fit together to form a map
f: * + M* tt* S o The projectivity of * + JIGG ~ of RO(G)-graded modules over z~ G . JIGG
follows immediately.
85
5. TH E MULTIPLICATIVE STRUCTURE OF t t~P(V) +. We assume that there are at least two distinct isomorphism classes of irreducibles in V; otherwise, the
. + mnltiplicative structure of l tGP(V ) is completely described in Examples 1.1.(h). As in section 3, we take 4) to be the set of irreducible summands of the complex representation V. Let 4)(0), 4)(1), (I)(2), ... be a proper filtration of 4). Then 4)(1) consists of exactly one representative of each of the isomorphism classes of irreducibles that appears in (I). Let ¢0, ¢1, ¢2, -.. , ¢,~ be an enumeration of the elements in 4)(1), and let n i be the number of elements of 4) isomorphic co ¢i (with n i ----(x> allowed). Arrange the enumeration of the elements of 4)(1) so that n o > n 1 _> ... _> nm. Extend the ordering of 4)(1) to (I) by selecting the unique proper ordering of 4) which is consistent with the filtration and in which, for each r > 1, the ordering of the representations in 4)(r+1) (l)(r) is the same as the ordering of the corresponding representations in dp(1). If the irreducibles which appear in 4) appear with equal multiplicity, then, regarded as an ordered set, • is a sequence of blocks, each of which is a copy of 4)(1). If the multiplicities are not equal, then 4) is still a sequence of blocks, but each block after the first will be either a copy of 4)(1) or of an initial segment of 4)(1). The lengths of the initial segments in the sequence can not increase. We will abuse notation by writing ¢i E 4 ) ( r + l ) - 4 ) ( r ) to mean that 4)(r+ 1)-4)( r ) contains an irreducible representation isomorphic to ¢i- We say that two sets of irreducible representations are equivalent if they contain the same number of irreducibles in each isomorphism class. Moreover, we sometimes identify equivalent sets of irreducibles.
Corollary 2.7 will be used to derive the multiplicative structure of H~P(V) + . +
from the multiplicative structures of I-IG(P(V ) )(e) and I-I~((P(v)G)+)(1). The
group I-I~(P(V)+)(e) is isomorphic to the nonequivariant cohomology group
HI~(P(V)+;7/), and we will think of the restriction map p as a map from
H~(P(V)+)(1) to KI~I(p(v)+;7/). Select an algebra generator x E tt2(P(V)+;7/) for
H*(P(V)+;7/). The fixed point space of P(V) is the disjoint union of the spaces
P(ni ¢i) ~ P(ni)- Let qi denote both the inclusion of the subspace P(ni) into P(V)
and the map H~(P(V)+)(1) * + --*I-IG(P(ni) )(1) induced by this inclusion. By
Examples 1.1.(h), * + HGP(ni) is a truncated polynomial algebra over H~S ° generated
2 + by an element x i in Hc(P(n i ) )(1). Let
c i: * + --* HG(P(ni) ) (1) /( torsion G im p)
denote the composition of qi and the projection onto the quotient. If y is in * + .
H~(P(ni)+)(1) , then [y] denotes its image in t tG(P(ni) ) (1) / ( tors ion ® i m p ) .
Throughout this section, we will index H~P(V) + on RSO(G) to simplify the selection of the integers d~. The comments in Remarks 4.11 on the passage from RSO(G)-grading to RO(G)-grading for * 0 I-IGS apply equally well to I-I~P(V) +. Recall that ,~ is the irreducible complex representation that sends the standard generator of
86
7]/p to e 2'~i/p and that ( is the real one-dimensional sign representation of 7//2. If p is 2, then 1, regarded as a real representation, is just 2~.
We begin with the case p = 2. Any complex irreducible representation is isomorphic to either the complex one-dimensional trivial representation or the complex one-dimensional sign representation A. Since P(V) and P(AV) are G-homeomorphic, we may assume that there are at least as many copies of the trivial representation in ~5 as there are copies of the sign representation. Thus, we may take ¢0 to be the trivial representation and ¢1 to be the sign representation.
* + * 0 By Theorem 3.1, t tGP(V ) , regarded as a module over ttGS , has one generator in each of the dimensions
2k + 2 k ~ and 2k + 2(k + 1)~,
for 0 < k < n 1 , and one in each of the dimensions
2k + 2 n 1 ( ,
for n I < k < n o . If one assumes n o = n 1 , or ignores the generators special to the case n o >111, then one might guess that, as an algebra, H~P(V) + had an exterior generator in dimension 2~ and a truncated polynomial generator in dimension 2(1 + ~). Except for the fact that the generator in dimension 2~ is not quite an exterior generator and for some difficulties in the higher dimensions when n o > nl, this guess is a good description of H~P(V) +. However, in order to describe the behavior in the higher dimensions as simply as possible, we adopt a notation that does not immediately suggest this.
r-r* S 0 H~P(V) + is generated by an T H E O R E M 5.1. (a) As an algebra over ~G ,
element c of H~(P(V)+)(1) in dimension 2 I and elements C(k) of H~(P(V)+)(1) in
d imens ions2k + 2min(k, n l ) ( , for l _ < k < n 0.
(b) For any positive integer k, let k denote the min imum of k and n I. Then the generators c and C(k) are uniquely determined by
0(e) = [0]
p(c) = x E H2(P(V)+; 7/)
and
; ( C ( k ) ) = x
Moreover,
87
and
2C + qo(c) = e x o e H G ( P ( n o ) ) ( 1 )
2~ + ql(C) ~-- e 2 -~" ~X 1 E H G ( P ( n l ) ) ( 1 )
qo(C(k)) = xok(e 2 + ~xo) i e t t~k+kt)(P(no)+)(1)
q](C(k)) = x~(e 2 + ~X1) ~ e H~(k+ki)(P(nl)+)(1 ).
n i If n i is finite, then x i = 0 and some of the terms in the last two sums above may vanish.
(c) The generators c and C(k) satisfy the relations
c 2 = e2c + ~C(1),
cO(k) = ~ C ( k + l ) , for k_>n, ,
and
C O + k ) , C@)C(k) = ?+~:_ ,~ . . ._
C ( j + k + i ) ,
fo r j + k _ < n l ,
fo r j + k > n I •
In these relations, we take C(i) to be zero if i >_ n o .
REMARKS 5.2. (a) By iteratively applying the third relation, we obtain
C(k) = (C(1)) k, for k _< n 1,
so that below the dimensions where we run short of copies of the sign representation, • p +
tIG (V) is generated by c and C(1). Moreover, in these dimensions, C(1) acts like a polynomial generator.
(b) If n 0 = n l , then H~P(V) + is generated by c and C(1). The only relations satisfied by these two generators are the relation
c 2 = e2c + ~C(1)
and, if n o < ec, the relation
C(1) n° = 0.
REMARKS 5.3. Notice that the maps q0 and ql behave differently on the generator
c. The element ~ = c + e 2 - n c of t tGP(V ) may be used as a generator in the
place of c and its behavior with respect to q0 and q1 is exactly the reverse of the behavior of c. To understand the geometric relation between these elements, observe that c and ~ can be detected in the cohomology of any subspace P(1 + A) of P(V) arising from an inclusion 1 + A C V. The space P(1 + A) is G-homeomorphic to S ~, but unlike S ~, it lacks a canonical basepoint. Either choice for the basepoint of P(1 + A) determines a splitting of I-I~P(1 + A) + into the direct sum of one copy of
8 8
• 0 • A I-IGS and one copy of PIGS . The canonical generator o f * A PIG S in dimension 2~ is identified with c by one of the two splittings and with ~ by the other.
, + When p is 2, the multiplicative structure of t tGP(V ) does not really exhibit
any complexities beyond those one might experience in a Z-graded ring. However, when p is odd, there are quirks in the multiplicative structure of H~P(V) + which are only possible because of the RSO(G)-grading. For tile odd prime case, recall the stairstep diagram obtained by plotting the dimensions a of the generators of I-I~P(V) + in terms of tctl and I~GI. Looking at this diagram in the special case where the irreducibles appearing in V appear with equal multiplicity, one might guess that I /~P(V) + was generated by two truncated polynomial generators, one in a dimension a with levi=2 and I a G t = 0 and one in a dimension ~ with Ifil=2m + 2 and lfiGl = 2. Unfortunately, such a guess would badly underestimate the complexity of I-I~P(V) +. The set of dimensions for a full set of additive generators must generate a larger additive subgroup of RSO(G) than can be accounted for by a pair of truncated polynomial generators. For example, recall that the first two additive generators of t t~P(V) + are in dimensions ¢~-1¢ 0 and ¢ ~ ( ¢ 0 + ¢1). If the additive generator in dimension ¢{1¢0 were to serve as a truncated polynomial generator, then the additive generator in the next higher dimension would need to be in dimension 2¢~-1¢0 instead of ¢~i(¢ 0 + ¢1). Any replacement of these two generators by an element and its square requires the introduction of further generators in some other dimensions inconsistent with a simple truncated polynomial structure. To provide a better feeling for the multiplicative structure of H~P(V) +, we give two sets of multiplicative generators. The first is a natural set with a great deal of symmetry. It does not exhibit a preference for any one ordering of ~. Unfortunately, this set is much too large. By selecting an ordering on ~, we are able to construct a much smaller, but very asymmetrical, set of algebra generators.
In order to describe the effect of the maps qi on our algebra generators, we must introduce more notation related to the integers d~.
DEFINITIONS 5.4. (a) For any two distinct integers i and j with 0_<i, j _< m, let
~i j denote the irreducible representation ¢}-1¢j, and let di-{ denote the integer d~, i j
for c~ = f i i j - f i r s . Note t h a t dij is 1 for any pair of distinct integers i and j. For any
integer i and any distinct pair of integers r and s such that 0 _ i, r, s _< m, let d/ / be
zero. The integers di.{ satisfy the relations
dirt d~; _= d / j mod p,
d~.{ + d¢~ _= d ~ mod p,
and
d~ j t~, t,, ~j dvw -= d ~ d,,~ rood p.
89
(b) If ¢i E t ( r + l ) - I ( r ) , then let c~i(r) denote the representation
¢ ~ 1 ~ ¢, and let ~l~j b e d o , f o r a = a i ( r ) - c ~ j ( r ) . Note that, i f ¢ i E i ( r + l ) - i ( r ) ,
d~i = 1. If either ¢i or Cj is not in i ( r + l ) - i ( r ) , then let d~j be zero. If ¢ i , ¢5'
and Ck are in I ( r + 1 ) - i ( r ) , then the integers d~j satisfy the relations
d~jd~k =_ dik mod p
and, if i :/= j,
d~'5 : (cti'J~r ik ak - - , - . 7 ~ ~ ( d j k ) m o d p, O<k<rn k ¢ i , j
where a k is the multiplicity of Ck in if(r).
THEOREM 5.5. (a) If i and j are distinct integers with 0 < i, j < m, then there is a unique element el5 in HZGiJ(P(V)+)(1) such that
I kJ jl, f o r 0 < k < m , ~k(cij) = di je~ i
and
P(eih) = x.
If r_>0 and ¢5 E I ( r + l ) - I ( r ) , then there is a unique element Ch(r ) in
I-I;J(~)(P(V)+)(1) such that
~'tk(Cj(r)) = e j%j (~ )_ rx , f o r 0 < k < m ,
and
p ( % ( r ) ) = x
The elements Cij , for 0 < i,j _< m and i ¢ j , and the elements Ck(r), for r_> 1 and * + * 0 Ck E i ( r + 1) - i ( r ) , generate HGP(V ) as an algebra over ItGS .
(b) For 0 < i , j , k < m and i :~ j ,
~j qk(cij) : dij fJ3ij "t- ~ ~ i j - 2 Xk"
(c) For r > 1 and Ck e I ( r + 1 ) - i ( r ) ,
90
1-[ (e&i + 2xk) 1 qk(Ck(r)) = x~ | ¢ i c #(r) ¢#/~i- •
If Cj e ~ ( r + 1 ) - ~ ( r ) and j ¢ k, then
~[,<i )~I I-I (d"e#ji+~#ji_2xk) l %(Cj(r)) = xkka~k co5 ~ + Ce~_ xk ¢i~#(~) k
r [ A k J ~ r ki ak~ - ~'vk) 1-I (d~) ¢i e ~(r) iSkj,k
%j(~)_~x~.
+
If ¢~ ~ ~ ( r + 1)-49(r), then qk(C/(r)) is zero.
(d) For 1 _<j _< m, let 7j be the representation ¢71 ¢i and let Dj be the i=0
j -1 element l-I cji in H;J(P(V)+)(1). Then the elements D j , for 1 <__j _< m, the elements
i=0 C0(r), for r>_l and ¢0 • ¢5(r+l)-qS(r) , and the elements DjCj( r ) , for 1"_>1 and
* p + n , S O ¢bj • ~ ( r + 1)-¢5(r), generate H G (V) as an algebra over . ,G .
REMARKS 5.6. In order to simpli~ our indexing, we define D O and C j(0), for
0_<j_<m, to be 1 C H~(P(V)+)(1). We also define 70 and oej(0) to be 0. Our
second set of generators for H~P(V) + is then just the set of elements DjCj( r ) , for
r _> 0 and Cj • ¢5(r+ 1 ) - q~(r). This set of elements of I-I~(P(V)+)(1) is also a set of • 0 additive generators of H~P(V) + as a module over t i cS . One might hope that a set
of multiplicative generators could be much smaller than a set of additive generators, but if the various irreducibles in • appear with very different multiplicities, then small sets of multiplicative generators do not exist.
We will order the set of generators Dj Cj( r ) by the dictionary order on r and then j. On the stairstep plot of the dimensions of these generators, moving in the direction of increasing order corresponds to moving up and to the right.
REMARKS 5.7. Nothing that has been said in the discussion of the odd prime case actually depends on p being odd; rather, mod 2 arithmetic is so simple that most of the technicalities necessary when p is odd are unnecessary when p = 2. The elements c and ~ in the case p = 2 are c10 and c01. The element C(j) is C0(j).
91
In order to describe the relations among the generators in H~P(V) + in a palatable form, we must introduce one more batch of elements in H~(P(V)+)(1).
DEFINITION 5.8. Observe that, for 1 _<j < m, tcDj is divisible by e.~j. Moreover, and
I -'d l o + qk(e~ ~Dj) = Pi~0 ji e HG(P(nkCk) ) / ( tors ionOim r). j - 1 j -1
Since l~ d~! is zero if k < j and 1 if k = j , the coefficients p I-t dk.! which appear in i = 0 jz i = 0 O~
the O k ( ~ ~Dj) form a matrix which is p times an upper triangular matrix with 1%
on the main diagonal. Applying the obvious analog of the process for diagonalizing
an upper triangular matrix to the elements e ~ D j
H~(P(V)+)(1) characterized by the conditions
and
p(k ) = 0,
( [p],
[ 0,
These elements can be described inductively by the equations
km ~ ¢-r I ~Dm
and, for l < j < m , kj = ¢-1 ~cDj m ~ I d k i
7j k=~j+l(i~O ji) ~k" m
Define k 0 ¢ H~(P(V)+)(1) to be ~ - E k j .
for j ¢ 0 then also characterize k 0 .
produces elements kj of
if k ----j,
otherwise.
The equations above characterizing kj j = l
Moreover,
f p, if k = j , q k ( k j )
0, otherwise.
kj Cj(r). These elements kj(r) are characterized by the equations H;J(~)(p(v)+)(1) For r_>l and Cj e • ( r + l ) - ¢ ( r ) , define &j(r)E to be
= 0,
and
p%5(~)_~x~ ,
O,
if k = j ,
otherwise.
Moreover,
92
qk(kJ (r)) = { p%j(,.)_~x~, i f k = j ,
0, otherwise.
For convenience, we define kj(0) to be k j . Observe that, for r > I, the elements kj(r) can also be constructed from the elements KDj Cj(r) in the same way that the elements kj are constructed from the ~cDj.
We begin our list of relations with the relation between any two of the Cij and the relation between any two of the Cj(r).
PROPOSITION 5.9. (a) Let i, j, r, and s be integers with 0_<i, j, r, s ~ m and i @ j , r ¢ s . Then
kj sj rs ks sj dij - dij - dij drs
Cij : (7flij-flrsCrs -~- dij ~Zij -1- E p ~[3ij kk" kCs
(b) Let r>_l and let i and j be integers such that ¢i and Cj are in • (r+l)-~(r). Then
-dkjdji Ci(r) = ~ i ( , - ) -~ j ( r ) Cj(r) -t- ~ p /~i(~)_~k(r ) kk(r ) .
kCj
An obvious initial response to this result is to assume that H~P(V) + can be generated as an algebra over H~S ° by any one of the cij and, for each r with q~(r+l ) -qS(r ) nonemepty, any one of the Cj(r). The k k and kk(r ) in the formulas spoil this simplification, especially since they are defined in terms of precisely the generators one would hope to omit. Solving this by taking the elements k k and kk(r ) as part of a generating set is hardly satisfactory since, from a Mackey functor point of view, these are torsion elements (because P(kk) and p(kk(r)) are zero).
The remaining results in this section describe the products of pairs of elements from either of the generating sets in terms of the smaller generating set. All of the relations in I-I~P(V) + follow from the relations in Proposition 5.9 and the relations below. If V is finite, then some of the elements appearing on the right hand
• V + side of these relations may not appear in the list of generators of It G (V) . Any such element is to be regarded as zero. We begin with the products which land in dimensions where there is no torsion. These are easily computed using the maps ~ and p.
PROPOSITION 5.10. (a) Let i, j, r, and s be integers with 0 < i , j , r , s _ < m and i @ j , r @ s . If I n > 2 , then
c i jcr ' = d0Jd0;~ lj is A0 j 0 s ) g -1310 C10 + a~D2 + ~j ~ij+/~rs -}- (d i j d r , ~ u i j drs Pij+~r s
kj k5 A0JA0~ [,41 j Is Oj Os kO kO kl - - drs)dl0 d-~ ~ d i j drs "-~ij "~rs -- k'~ij drs d i j - d20 d21
k=2 P e~3i j+flrs kk ,
93
where a = f l i t + ~ - 72"
If m = l , then
Oj O~ eij Crs = di j dr~ f~ij+~r s
Co(l). ~S i j+J3r~-OeO(1)
(b) Let i, j, and r be integers with- 0 _< i, j < m, i @j, and
( , l j 1~ ,Oj ,0s, -Jr- ~ d i j d r s - d i j dr~)e/3i j+~rs_fl loclO q-
rj e i j Dr = d i j ~'~.. D~ + o-~ D~+ t +
~3
(dkJ drj)rI:[1 "k~ r dk ~ m ,--ij - - - - i j , xxdrs - d - a 1-I r+l,s
E s=0 s=0 p k=r+l
l < r < m . Then
^
¢ ~ij+Tr ~k '
where ~ = t3ij + 7,- - 7~+a.
(e) Let i, j be integers with 0 _< i, j _< m and i =)kj. Then
"~J Dm + { Co(1 ). ei j Dm = d i j £~ij flij+'~rn--aO(1 )
(d) Let i , j , r, and s be integers with 0_< i , j , s_<m, i ¢ j , r_>l, and ¢~ E O ( r + l ) - ~ ( r ) . I f¢1 E ~ ( r + l ) - d p ( r ) , t h e n
Oj ~r e i jC, ( r ) = di j do, e&j+~,(~)_~o(~)Co(r) + a ~ D 1Cl(r ) +
k j ~ r _,.tOJ~ir r AkO~Ir d-cr ~] d i j d k , "*ij=o, d k o - ~ l o ~ k l P k>l
Ck e ~(r+l)-~(r) e ~ij+as(r)-ak(r ) kk(r),
where a = f l i j + a s ( r ) - 7 1 - %(r).
If 81 ~ ~(r + 1 ) - ~i'(r), then Oj ~r
e i jC . ( r ) = d i j d o , e&jCo(r ) + 4&j+a0(r)_~0(r+l)C0(r+l ).
(e) Let i , j , r , a n d s be integers with 0_< i , j , s_<m, ig : j , 8~ e ¢ 5 ( r + l ) - ~ ( r ) . If ¢,+1 E ~ ( r + l ) - e ( r ) , then
cij D, C~(r) 'J = di j epi j D, Cs(r) q- ere, Ds+ 1Cs+l(r) q-
dk.s+ 1 d_a I~I d kt Z a L (d:} - d:~)t*=Iff°d:: - - ~ t=O s+l,t
P k>s+l
r_> 1, and
% ~k(r),
94
where c ~ = / 3 i j + 7 ~ + o ~ , ( r ) - 7 , + l - c % + l ( r ) and 6 k = / 3 i j + 7 " + ~ , ( r ) - c ~ k ( r ).
If ¢s+1 ff ~(r -t- 1) - ~(r), then
*J Cs(r) + ~/3ij+.rs+as(r)_c~o(r+l)Co(r+l ). Cij D , C s ( r ) : dij ePijDs
(f) Let r, s_> 1 and assume that i < j _< m. O(r + s) appear with equal multiplicities, then
~ r ~ S . ~ r + s
%(r)%(s) = %(r+s) + E dkjdk - p Ck e q~(r+s+l )--q~(r+s)
If the irreducibles that appear in
Poj(r+,)-ak(r+, ) kk(r +s)"
Moreover, the integers "~kj may be selected to be the products dk jdk j so that the kk(r + s) correction terms are not needed.
Since the elements kk(r ) appear in so many formulas, we include a description of products involving them.
LEMMA 5.11. Let i, j, k, r, and s be integers with 0_<i,j ,k_<m, ¢k E ~ ( s + 1 ) -~ ( s ) .
(a) If i C j, then kj
cijkk(s ) = dij e&jkk(s ).
r, s>O, and
and
(b) If ¢j E ¢(r + 1) - ~(r) and ¢~ E ~(r + s + 1) - qS(r + s), then
Dj Cj(r) kk(s ) = kj t~=odj e T j + o j ( r ) + ok(s ) - ok(r+s ) ;¢k(r+s)"
In the formula for Cj(r) kk(s), replace %j(r) + %(s) - ok(~+s ) by
#oj(r) + % ( , ) - %(~+s) if ]aj(r) + C%(S) - ak(r+s)] is zero.
(c) If ¢j C ¢(r + 1 ) - ¢ ( r ) and ¢k 6 ¢(r + s + 1 ) - ¢ ( r +s ) , then Cj(r) kk(s )
and Dj Cj(r) kk(s ) are zero.
To complete our description of the multiplicative structure of HOP(V) + we
need to describe the products of various pairs made from elements of the types Ci(r),
95
DjCj( r ) , and D k, If we use the convention that D O = Cj(O) = 1, then the products
we must describe are all special cases of the general product (Di, Ci( r ) ) (Dj , Cj(s)),
where r , s_>0, ¢i E ~ ( r + l ) - ~ ( r ) , ¢j C ~ ( s + l ) - ~ ( s ) , i' is 0 or i, a n d j ' is 0 or
j. We may assume that i' > j ' . Recall the formula given in Theorem 5.1(c) for the
product C(j)C(k) when p = 2 and j + k > n I. Observe that this formula may be
o b t a i n e d from the binomial expansion of (¢ 2 + ~x) j+k-nl by replacing the powers of
x by various generators C(t). The formula for our general product is related in a
similar way to the expansion of an expression of the form l~I (ai + b/x). The i r a0
summands in this expansion are indexed on the subsets of the set {0, 1, ... ,n}. The
summand corresponding to the subset I is
( l ~ a i ) ( ] ~ b i ) x [I[, i~I i~I
where ]I] denotes the number of elements in I. To describe the analogous part of our
formula for (Di, Ci(r))(D j,Cj(s)), we must specify the indexing set which replaces
{0, 1 . . . . ,n}, the factors which replace y [ a i and l~b i , and the procedure for
replacing the powers of x by the appropriate D~ Ck(t ).
In the p = 2 case, describing how the powers of x are to be replaced by the
generators C(j) is very simple because, if j > n~, then the next generator after C(j) is
always C(j + 1). However, when p is odd, the generator after D k Ck(r ) may be either
Dk+ 1Ck+l(r ) or C0(r+ 1). To handle this complication, we introduce two functions f
and g from the nonnegative integers to the nonnegative integers. These functions are
to be chosen so that, for any i_> 0, Cf(i+l)(g(i+ 1)) is the generator immediately
following Cf(i)(g(i)) in our stairstep ordering. If Cf(,~)(g(n)) is the last generator in • ~ + H~P( ) , then we define f ( i ) = 0 and g ( i ) = g ( n ) + i - n for i > n and use the
convention that Dj Cj(r) is to be regarded as zero if it does not appear in the list of
generators of I I~P(~) +. Each time we use this notation, the initial values, f(0) and
g(0), of the functions will be specified to suit the particular application.
The indexing set which replaces the set {0, 1 . . . . . n} is related to the
difference in dimension between the product (Di, Ci(r))(Dj, Cj(s)) and the lowest
dimensional generator D i , C i ( r + s ) which should appear in its description. If r_> 0
96
and 0 < j _< m, then define the subset ~j(r) of d2(r+ 1) by
(I)j(r) = ~(r) U {¢i: i < j and ¢i E ~ ( r + l ) - ~ ( r ) } .
Let ~i,(r) U ¢sj,(s) denote the disjoint union of the sets ~i,(r) and ~/,(s). Our
replacement for the set {0, 1 . . . . ,n} is the set • obtained by deleting from
• i,(r) U ~j,(s) a subset equivalent to the set (I)i,(r+s). We abuse notation by
writing ~ as ~i,(r) 1_1 ~ j , ( s ) - ( I ) i , ( r+s ). Observe that ~j,(s) is equivalent to the
disjoint union of • and ~ i , ( r + s ) - ¢5i,(r). Let u be l ~ l - 1 and number the elements
of ~ from 0 to u. Let h be a function from the set {0, 1 , . . . , u } to the set
{0, 1 . . . . , m} such that the i th element of • is isomorphic to the irreducible
representation q ) h ( i ) "
One of the coefficients appearing in our formula is determined by a certain
element a of RSO(G) with Ic~l = 0 and tc,~I _< 0. This coefficient will be go if
Ic~GI < 0 or c o if laGI = 0. To simplify our notation, we write )~o for either of these,
relying on Ic~cl to indicate whether ~ or ao is intended. Another coefficient will
depend on a certain element f3 of RSO(G) with 19 I- 0 and I~31 _> o. This coefficient
will be e5 if I/3I > 0 a n d / ~ if It31 = 0. We write 0~ for either of these, relying on 1/31
to indicate which is intended.
PROPOSITION 5.12. Let i, i', j, j ' , r; and s be integers with r,s_>0,
¢ i E ~ ( r + l ) - q S ( r ) , C j E ~ ( s + l ) ~(s), i ' = 0 or i, j ' = 0 or j, and i ' > j ' . Let
= ¢5i,(r ) t_J ~j,(s) - (I)v(r + s ). Initialize the functions f and g by
and
i', i f¢ i , E ¢ ( r + s + l ) , f(0) =
0, otherwise,
r+s , i f¢ i , C ( I ) ( r+s+l ) ,
g(0) = r + s + i, otherwise.
Let u : l q * l - 1 and number the elements of 9 from 0 t o u . Let A C 9 and let s ' a n d
97
s" be the number of elements isomorphic to Cj in A and q~i , ( r+s ) -~ i , ( r ) ,
respectively. If the subset A of • contains the elements numbered J0, J l , ... , j~o,
with J0 < J l < . . . <J~ , then let
and
I df(Jt-~)'h(Jd~ I df(J~-t)'J~ d~ = t~=o j ,h ( j t )~ t~=o jk ~, h ( j t ) ¢ j h ( J t )=J
t = 0 j , h ( j t t = 0 /3j h ( j t ) ~ j h ( J t ) = J
where
Xz~ Xa ,
a = c - i F ~ ' 1
Lh( j t )7~ j C t ~ O i¢( r + s ) - ~ ir( r
¢ i Ct eOi~(r)
e ~f(l AI)(g(IAI))
The tag j :/= 0
present only if j ~ 0.
trivial representation.
on the bracket about the (s' + s " ) ¢ j l ¢ 0 indicates that this term is
The 2s term in a indicates 2s copies of the real one-dimensional
If a E RSO0(G), then let.
d a = d o.
I f A = ~ , t h e n l e t d~x, e ~ , ( t , a , a n d X be 1. A
where
I f i ' < k < m a n d ¢ ~ E ~ ( r + s + l ) - ~ ( r + s ) , l e t
O k = 0/~,
98
= ~'dr) + %.(s) + -y~, + L , - ~ , ( r + s ) ,
and let A k be
1 ~ i ~ i ' -1 kJ(t~=od~tt)( I-I dkt ~ -" ~ [-'~r+s(v-I kt\ 0 rid.) E - t=0 J'*] v=i' t=0 lal=v-i'ac~'
Then
0 0 Cf(taf)(g(1Al)) (Di, Ci ( r ) ) (Dj , Cj(s)) = ~ d~,_~ ee-za X Df(Iz~f) ~ c ~ z~
Ak 0k kk( r+s) . k=i I
+
REMARKS 5.13. (a) Let r_> 1. If d2(r) contains r copies of every irreducible complex G-representation, then (~i(r) is independent of i and it is easy to see that C i ( r ) = C j ( r ) for every i and j such that ¢ i , 4~j E ~ ( r + l ) - q ) ( r ) . Moreover, Cj(r) = C j(1) r. Thus, if ~ contains every irreducible complex G-representation and these representations appear with equal multiplicities in ~, then Ci(r) generates a polynomial, or truncated polynomial, subalgebra of H~P(O) +. In this case, the elements Dj , for 1 _<j _< m, and Ci(1), for any i, generate I-I~P((I)) + as an algebra
* 0 over HaS .
(b) If p = 3, then we may choose the integers d , so that da = +1 for every a in RSO0(G ). When this is done, the assignment of d~ to a is a homomorphism fl'om the additive group of RSO0(G ) to the multiplicative group {5:1}. With this
choice of the integers d~, all the relations among the d~ j and the d~;j given in
Definitions 5.4, except the one involving a sum, hold in 7/as well as in g/3. If r _> 1 and ¢i , Cj E ~ ( r + 1), then
CAr) = %d~) -~¢~) C~(r).
Thus, the only elements of the form Cj(r) needed to generate H~P((I)) + as an algebra over H~S ° are the elements C0(r ) for r > 1. Also, a pair of elements cij and c~, will generate D 1 and D 2 if (tk(cijcr~) is nonzero for only one value of k. In particular, c01 and %o generate D 1 and Du. When all three irreducible complex G-representations of 7//3 appear in • with equal multiplicities, c01 , c~0 , and C0(1 ) generate I-I~P(~) + as an algebra over I t , S °.
6. PROOFS. Tile results stated in section 5 are proved here. As indicated in Remark 5.7, our results for p = 2 are a special case of the results asserted for odd
9 9
primes. They have been presented separately only because they can be stated so simply. The proofs given here are independent of whether p is 2 or odd. We begin by construct the elements c~j and Cj(r). We then show that they generate H~P(V) +
• 0 as an algebra over H~S . Finally, the relations stated at the end of section 5 are verified. Throughout this section, • is a set of irreducible complex representations of 7//p and ~(0), ~(1) . . . . is a proper filtration of ~. We order the elements of q~ in the standard proper ordering introduced in section 5. Recall the maps qi and Cti and the cohomology classes x and x i from the introductory remarks in section 5 and the representations a~(r), /3~j, and 7j from Definitions 5.4 and Theorem 5.5(d). If A C q, then x also denotes the image of x E t t~(P(¢)+)(e) in H~(P(A)+)(e); thus, the powers of x are thought of as the standard additive generators for the nonequivariant cohomology of all the sub-projective spaces of P(q) . For each integer j with 0_<j < m , let pj(4~) be the component of the fixed point space of P(¢) associated to the irreducible representation 6 j .
The classes cij and Cj(r) are constructed by defining them on the smallest possible projective space and then inductively lifting them to larger projective spaces.
CONSTRUCTION 6.1. (a) Let i and j be distinct integers with 0_< i, j _< m. The
space P({¢j}) is just a point and the space P({¢i , C j}) is G-homeomorphic to S #ij. The inclusion of P({¢j}) into P({¢i , ¢j}) induces the cofibre sequence
p({¢j})+ q4 5 s ei . ~ i j fig *
Let ciy ~ ttG (P({¢g, ¢i})+)(1) be the image of 1 e A(1) ~ H iJ(sZiJ)(1) under ~r .
Then q j ( c ~ j ) = 0 by exactness and q~(cij ) = e#ij by the commutativity of the diagram
p({qSi})+ qi , p({q~i,q~j})+
S O efliJ~ 813ij.
These are the correct values for qi(cij) and qj(cij ) because x i and xj are zero. Since
the map 7r*: H~iJ(S#iJ)(e).-, H~iJ(P({6i ,6j})+)(e) is an isomorphism in dimension
/3~j, p(c~j) = x.
Let ~ be a subset of (I, which properly contains the set {¢i, ¢j} and assume that, for every proper subset A of • containing {6 i ,6 j} , cij has been defined in
HGiJ(P(A)+)(1) and has the proper images under the maps qk and p. Pick an
irreducible representation 6t which appears in • at least as often as any other irreducible. If no irreducible appears more than once in tit, then we may also insist
100
that t ¢ i , j. Let Z X = ~ - { ¢ t } , and let V be the representation ~b~-lEqS.
inclusion of A into • induces the cofibre sequences
The
p(A)+ 0 p ( , ) + -~ S v
and
Pt(A) + ~ P,(•)+ -~ S VG.
We will lift the class cij ¢ H~iJ(P(A)+)(1) along the map
e*(1): It G (P(~)+)(1) -, H~iS(P(A)+)(1)
induced by 0. To distinguish the class ci5 and its lifting, we will denote the class in r ~
H~'J(P(&)+)(1) by cij. The maps qk, for k 5Lt, factor through 0"(1), so any lifting
of cij along 0"(1) will have tile right image under q~, for k ~ t. Moreover, since 0*(e) is an isomorphism in dimension /3ij , any lifting of aij will also have tile right image under p.
It remains to show that we can choose a lifting of cij with the correct image under qt . We have chosen t so that the long exact cohomology sequences associated
to our cofibre sequences have zero boundary maps. If IvGt > 2, then HGiS(sV)(1)= 0
and we take cij to be the unique lifting of e O. If IVGt > 2 , then 0 t induces a cohomology isomorphism in dimension /3ij and this lifting of cij along 0"(1) must have the correct image under q,. If IvGI = 2, then the short exact, sequence
splits. The end terms are
tt "s ~ at " (zx) + ~ = R and Pt = •
The image of 1 ¢ ?7 = R(1) in H ~ J P t ( ~ ) + is {&j_2xt. By our induction hypothesis,
tj O;(1)qt(cij) = qt(aij) = dij e&j.
Since P ( c i j ) = x , pq,(ei j) is the generator of H~iJ(P,(O)+)(e). It follows ttlat tj
q t ( c i j ) = dij e~i j + ~i3ij_2 x t "
If IvGI = 0, then no irreducible appears more than once in ~ and we have selected qS, so that t :/: i, j. In the diagram
L ev I q~ ~ q,
o - . + o
-~ 0
I0t
comparing the cohomology sequences of our two cofibre sequences, we have that
I-I~ ijS V and tt~ ijS ° are (g) and the map e V is multiplication by p. Thus, if z is a
lifting of ci j , then by adding elements from the image of I-I~GiJs V to Z, we can adjust
qt(z) by any multiple of p. It now suffices to show that there is a lifting z with
tj mod p. The lifting problems for P(g/) and P({¢i , Cj, Ct}) can be q~(z) ~ dij e~i j compared via the cohomology maps induced by the inclusion of {¢i, Cj, Ct} into ~.
This comparison indicates that it suffices to show that the lifting problem can be
solved when k0 = {¢i, C j , Ct}. In this case, consider the diagram
0 --+ H P G i J s V /3 i j + O* "" H G P ( ~ ) -~ l t ~ ' e ( z x ) + -+ 0
l e [ q ~qj
0 -~ It~iJs ~tj 7 ; i j e -* tt G })+ -+ H ({¢d' Ct})+ qj i jp({¢j -'+ 0
comparing the cohomology exact sequences for the pairs (p(ko), p(A)) and
(P({¢ j ,¢ t} ) , P({¢j})). Let a = (3ij - /3 t j . If z is a lifting of eij along 0"(1), then _~_ LI fl i J [ ~ fl qj(z) q j q ( z ) = 0 . Thus, q ( z ) = 7 ( y ) for some y e .~G ,~" tJ)(1) ' Since pq(z) is
the generator x of H~ (P({¢j,¢¢})+)(e), p(y) must generate H~iJs&J(e), and y
must be a~ + na~ for some integer n. The diagram
G
*e -[qt
commutes and gives that qt(z) = qqt(z) = e(y) _= e(c~) mod p. By the definition of tj o'~,, e(o'~) = dij e&j.
(b) Let r_> 1 and let Cj E q~(r+l). The cofibre sequence associated to the inclusion of P(~(r)) into P(O(r) W {¢~}) is
P(~(r)) + -* P(~(r) U {¢j})+ -~ S ~'j(').
Define C j ( r ) e t t G (P((I)(r)U{¢j})+)(1) to be the image under ~*(1) of
1 E A ( 1 ) = H Since Tr* is an isomorphism in dimension c~j(r),
p(Cj(r)) = x The cohomology diagram in dimension c~j(r) induced by the
diagram
102
Pj(¢(r) u {%})+ 7rj
S~
qJ P(@(r) U {¢j})+
l- e ~j(~)
---* S
indicates that qj(Cj(r))=ec, j(r)_~x~j. If k@j , q k ( C j ( r ) ) = 0 for dimensional
reasons. As we did with the definition of cij in part (a), we extend the definition of Cj(r) to H~P(@) + by working inductively along a sequence of subsets of ¢5 between ~5(r) U {¢j} and ~. The only difference between the argument given for cij and the one which should be used for Cj(r) is that the liftings of Cj(r) should be chosen to behave properly with respect to p and ~1~ instead of p and qk. This change is necessary because %(Cj( r ) ) is more complicated than qk(cij). The behavior of the Cj(r) with respect to the maps qk is established in the lemma below.
LEMMA 6.2. Let r _ > l a n d ¢~ E d 2 ( r + l ) - ~ ( r ) . Then
r q/c(Ck(r)) = X k 1-I (~9~, + )]
¢i e q~(r) ~ fiki - 2 x k •
i¢:k
If Cj e ~ ( r + 1)-4p(r) and j # k, then
r d ~i e = + . %,_
' k ~¢j,k
+
r [dkJ~ r ki 21 ~kj ~ jk)
i=/=j,k
%j(~)_~x~.
If Ck ~ q?(r+ 1) q)(r), then qk(Cj( r ) ) i s zero.
PROOF. If ¢~ ~ 4p(r+ 1) - qS(r), then %(Cj( r ) ) vanishes for dimensional reasons. Therefore, assume that Cj, Ck ¢ ~5(r+l)-dp(r) . Let
= ¢(r) u {¢: ¢ • ¢ ¢(r) and ¢ ~- Ck}"
The {mage of the class Cj(r) in II;P(dp) + under the map
H~P(¢~) + + ItSP(k~ U {¢j})+
may be computed using the maps p and ~!i- It is the class Cj(r) in t t ~ P ( ~ U {¢j})+. The image of this class under the map
since Pk(~) = Pk(9) and the map qk for P(~) factors as the composite of the map
I t~P(~) + -* I-I~P(9) + and the map qk for 9. Observe that
(r~J (~)-~k(~) ---- (crPjk-Pkj ¢i t~c~ j ( r ) - ~ k ( r )
for some integer a. With this description of O~c~j(r)_c~k(r), it is easy to derive the
formula for qk(Cj(r)) from the formula, for qk(Ck(r)). The formula for qk(Ck(r) is
derived using an iterative procedure. Let s > r and pick Ct E • with t :~k. The
image of Ck(s ) E I-I~(P(~)+)(1) under the map I t~P(~) + --* I-I~P(~ - {¢~})+ is
czktCk(s ) + {Zkt_2Ck(s+ 1).
Iterating this process to eliminate from q~ all the irreducible representations not isomorphic to Ck, we move from H~P(~) + to It~P(n~ ¢~)+ ~H~P~(gl ) + and from C~(r) to the expansion of
(%,+ t [¢i ~ ~(r) ~ i -
On the other hand, the image of Ce(r) under this sequence of transformations must be q~(Ck(r)).
Now that we have defined the classes cij and Cj(r), we must show that they • 0 generate H~P(¢) + as an algebra over ttGS .
PROPOSITION 6.3. The classes cij , for ¢ i , Cj E (I)(1), and the classes Cj(r), for
r > 1 arid Cj C ~(1" + 1) - ~(r), generate H~P((P) + as an algebra over H~S °.
PROOF. If • is infinite, then, by the proof of Theorem 2.6, H~P(~) + is the limit of the ~ P ( A ) + where A runs over the finite subsets of ~. Thus, it suffices to prove the result for • finite. Recall the functions f and g and the subsets ~j(r) of defined in the remarks preceding Proposition 5.12. For this proof, initialize f and g by f(0) = 0 and g(0) = 0. We will show, by induction on n, that the classes cij and
, + Cj(r) which are defined in HGP(~f(,~)(g(n)) ) generate that Mackey functor as an
• 0 algebra over t t G S . The result is obvious for n = 1, since (I)f(1)(g(1))= {¢0} and
104
P({¢50} ) is a point. Assume the result, for n. Denote c~f(n+l)(g(n+l)) + 7f(n+l) by
ct. The boundary map is zero in the cohomology long exact sequence associated to
. + All of the classes ci# and Cj(r) which are defined in ItGP(¢f(,~)(g(n))) are also
defined in * + HGP(~f(n+l)(g(n+l)) ) . Moreover, 0* takes these classes in . +
H~P(<I)f(n+l)(g(n+l))) + to the corresponding classes in HGP(¢f(,~)(g(n)) ) . Thus, to • + * 0 generate HcP(45f(,~+~)(g(n+l)) ) as an algebra over HGS , it suffices to add to these
classes the image z of the canonical generator of A ( 1 ) = H~(S~)(1). Clearly, p(z) is O' + the generator of HG(P(~f(=+l)(g(n+l)) ) )(e). Moreover, for k : f i f (n+l ) , ~ k ( z ) = 0
since qk factors through H~P(<l)f(,~)(g(n))) +. Finally,
commutes. The elements z and Df(n+l)Cf(n+u(g(n+l)) must be equal since they
have the same image under the maps qk and p.
The equations in Propositions 5.9 and 5.10 describe elements in dimensions where there is no torsion. As a result, these equations can be checked easily by applying the maps p and qk to both sides. The equations in Lemma 5.11 are easily checked using the maps # and qk because the images of the classes kj(r) under the maps qk are so simple. However, the formula in Proposition 5.12 is more difficult to verify.
PROOF OF PROPOSITION 5.12. We may assume that l~I _> ]~i,(r) I + qsj,(s) so
105
that all of the Df(lal)Cf(iz~l)(g(lAI)) on the right hand side of the equation are
nonzero. If I¢1 is too small, then form a sufficiently large set ~ ' by adding enough copies of ¢0 to ~. The proof below applies to 4p'; the result for ~ is obtained using the cohomotogy map induced by the inclusion of • into ~' . We show the equality of the images of the two sides of the equation under the maps p and q~. Since the map p preserves products, p(D i, C i ( r ) D , Cj(s)) is the generator of H~(P((IS)+)(e) in the appropriate dimension. The only term on the right hand side of the equation in Proposition 5.12 which is not in the kernel of p is the summand corresponding to
regarded as a subset of itself. This term is X Df(u)Cff~)(g(u)) and its image under p
is the generator of H~(P((I))+)(e) in the same dimension. Thus, the expressions on the two sides of the equation have the same image under p.
Let k be an integer with 0 < k < m . If Ck ~ ~ ( r + s + 1 ) - <I)(r+s), then both sides of the equation vanish under % . If Ck C (I ) ( r+s+ 1) <I)(r+s), then expand the polynomial obtained by applying qk to Di, Ci(r)Dj, Cj(s ). Each term in the expansion consists of the product of an integer, a power of x k , and an element of the
, HaS . We classify these terms according to the factor form ~ , ~ or e ~ from * 0
from . 0 HaS . There is exactly one term with a ~ ; its integer coefficient is one. There
is exactly one term with an c~; its integer coefficient may be zero. This term is
exactly the part of qk which is detected by qk. There may be any number, including zero, of terms containing a product e~ ~ . These terms are all torsion elements of order p.
Expand the polynomial obtained by applying qk to the right hand side of the equation and observe that the same three types of terms appear. The su mma n d
indexed on q~ regarded as a subset of itself is the only source of a ~ . It is easy to
see that this ~ term exactly matches the corresponding term from the left hand side of the equation. If i ' > k, then the expansion of the image of the right hand side under qk will contain no ~ term. In this case, ~k(Di~) is zero and the image of the left hand side under qk also lacks an ~ term. If i ' < k, then numerous summands contribute to the c~ term of the left hand side, but the coefficient of the k k ( r + s ) term is explicitly designed to ensure that the c z terms of the expansions of both sides match. The only problem here is that it is not obvious that the coefficient A k of k k ( r + s ) is an integer. To show that A k is an integer, it suffices to show that, modulo p, the image under qk of the left hand side is equal to the image of the part of the right hand side indexed on the subsets of ~. Since the e ~ terms are all
O~. torsion of order p and the k t ( r + s ) summands on the right hand side contribute nothing to them, proving the equation
q~ }--~ d~,_A ~ _ A xADf(i,al)Cf(l~l)(g(]A]))) m o d p qk(Di, Ci(r) D j, Cj(s)) -- (~,c,,
also shows that the c~ ~ terms of the two sides agree and so completes the proof of the proposition.
We prove this equation modulo p by transforming the right hand side into
106
the left. In Theorem 5.5(c), %(Cj ( r ) ) is described as a sum of two terms when j ¢ k. The second term can be ignored in this t ransformation process because it vanishes
modulo p. Recall that each * a is a ~ , for some virtual representation a'. We
accomplish our transformation by writing a as a sum of differences r / - ¢ of
irreducible complex representations. We then rewrite X a = Xa as the product of the
elements X _ ¢ . To see that such a rewriting is justified, recall that if /3 and 7 in
RSO(G) are chosen so that the elements below are defined, then in H~(S°)(1)
( ~ 7 = {0+y {~ % = 0 e 0 % = pc0+ 7
and
(r 0 o " = ~ 0 + ~ + A~0+~,
where A is some integer depending on /3 and 7. Now observe that every summand in
the expansion of %(Df(l~l)Cf(lai)(g(IAt)) ) contains either an c a or a ~ . Thus, the
as the product of the X ~ + ~ error terms that might arise in the rewriting of Xa ~-¢
are killed by the e~ and ~ from %(Df(l,~t ) Cf(lz~l)(g(lAt)) ).
We perform our transformation of the left hand side in four stages. During the first three stages, we think of the left hand side as a sum indexed on the subsets of q* and work on each summand separately. Therefore, fix a subset A of • and let
a be the virtual representation such that ;(a = )~" Recall that s' and s" are the
number of elements isomorphic to ¢j in A and ~ , ( r + s ) ~ , ( r ) , respectively. Recall
that u = Ig'l- 1, that the elements of q~ are numbered from 0 to u, and that h is a function from the set {0, 1 . . . . . u} to the set {0, 1 . . . . . m} such that the i th element in • is isomorphic to Oh(i)" Assume that the elements of • numbered J0, J l , --- , jw, with J0 < J l < - - - < J * ~ , are in A and that the elements numbered i0, i 1 . . . . , i~, with i 0 < i 1 < . . . < i ~ , are in g~ -A . For any integers q and t, with 0_<q, t_<m, abbreviate e~q t and ~ q t _ 2 by eqt and 4q~. Define the elements a l ,
~f(lal/~ _lf(Izll) ¢ k k i' >f(l~l)> k or f(I~l) > k _> i'
. . . . . d ~ e k factor. Observe that the (1S ~ e S a factor has been transformed into a ~ - ~ ~ - a
This is accomplished by the [ ( s - s ' s " ) (4 ;71% ¢)-1 ~50)]0 # j,k summand in a~. If
k = 0, then obviousiy no such transformation is needed. If j = 0, then there will not
k k will not depend be any elements of 9 isomorphic to 4D j , and the value of d ~ _ a e e _ a
on k. In the description of the factor above indexed on t, for 0 < t < w, and
throughout the third stage of the transformation, the set Ofd~[)(g(IA[)) - (I)i,(r 4- s) is
109
identified with the set {¢f(t) : 0 < t < W). By this identification, constructions that
would naturally be indexed on (I)f(t~l)(g(]AI))-¢i,(r + s) may be indexed on t. The
description of the set {¢f(t) : 0 < t < w} involves our usual abuse of notation in that,
whenever q ¢ t and f(q) = f(t), the representations el(q) and ef(t) are intended to be
distinct, but isomorphic, elements of the set.
The factor
qk(Di ' C~(r)) x~ I¢t ]-I (d kt
¢¢it(ra'+~s)-q~it(r)\ jt (jr tJj,k
"4- ~ jt Xk djk 6jk $ t t
appears in every summand of the transformation of the right hand side of the
equation. We therefore factor it out of the sum and ignore it for the rest of the
transformation. Observe that this factor consists of qk(Di, Ci(r)) and that part of \ /
% ( D / C j ( s ) ) which is associated with the set ~ i , ( r+s ) - ( I ) i , ( r ) when q) i , ( s ) i s
regarded as the disjoint union of • and ~ i , ( r+s ) -ep i , ( r ) . Thus, we must transform
what remains of the sum after this factor is removed into the part of q ~ ( ' ~ , % I s ) )
coining from ~.
In the third stage of the transformation, X 3 is used to transform the
remaining part of the A summand into
{, x,) n + ~'-z.4 q~-A I t~3 \ j,h(jt) j,h(jt) ~- Lh(57)~5 J,,(h) _] Lh (}7)o= j
For the fourth stage of the transformation, consider the subsets A of ~ that
contain the last element eh(u) of ql. The summands indexed on A and A-{¢h (u )}
contain the common factor
h(~t)O j Lh (: =)~: j j,h(it: ~(i~)=j j
110
te--i dk,f(t ) ) i w-i ~,f(t) ) l
Lh(J?i#j J'h(Jt) Lh(Jt)=j
which we have written down using the i, and Jt numbering of the elements in ~ - A
and A. Each of the two summands contains exactly one term not in this common
factor. If h(u) ¢ j , then these terms are
= d k,h(u) dfj(hg'~(*')ej,h(,~ ) + d k ' ~ ) e h~ + x~. , J, ( ) J, ( ) ~j,h(u)Xk j,h(u) ej,h(u) 4- ~j,h(u)
If h(u) = j , then these terms are
df(~),/ dk,f(w) k,j j,k e l k + j,k e lk + ~j,k xk = dj,~ ej,k + ~j,k xk"
In either case, the result is independent of A and may be factored out of the sum.
Moreover, this factor is exactly the contribution that Ch(~) should make to
%(, Dj, C/(s))~ when Ch(~) is regarded as an element of Cj,(s) under the identification %
of Cj,(s) with the disjoint union of • and ~i,(r + s ) - ~pi,(r).
The sum that remains after the factor associated to Ch(~) is removed may be
regarded as one indexed on the subsets A of * - {¢h(~)}" We now pair the summand
indexed on a subset A containing the last element Ch(~-i) of q - { ¢ h ( ~ ) } with the
sumrnand indexed on A - { ¢ h ( ~ _ i ) } to obtain the factor of % ( D j , Cj(s) ) associated %
%
to Oh(u-i)" Repeating this process until the elements of ~ are exhausted, we recover
the part of q JD\ j, Cj(s)]] associated with kl/.
APPENDIX. Computing H~S °. Here, we outline the calculation of * o HGS . The computation of the additive structure and, for G = 7//2 or 7]/3, the computation of the multiplicative structure are unpublished work of Stong.
Three cofibre sequences suffice for the computation of the additive structure of H~(S°). Recall that ( is the real 1-dimensional sign representation of 7//2. Let r/ be a nontrivial irreducible complex representation of G = Z/p, for any prime p. Let G+--* Sr] + be the inclusion of an orbit and let Srj+~ S O and S~+~ S O be the maps collapsing the unit spheres Sr] and S( to the non-basepoint in S o . The cofibre sequences associated to these maps are
G + ~ St/+ -, EG +
S + -, S O e S ~
111
and
G + ~ S¢+_+S O e_, S("
The first step in the computation is obtaining the values of H,GSrj + and * +
H GSq from the first cofibre sequence.
LEMMA A.1. For any nontrivial irreducible complex representation rl of G,
t L, g _ ,
t t 0 s ~ + = < a ,
R_,
0,
iflc~l = 0 a n d ] G]iseven,
if lal = 0 and ]c~ G] is odd,
if Ic~l = 1 and ]e~GI is odd,
if lal = 1 and la G ] is even,
otherwise,
t t ~ S v + =
"R,
R_,
L,
L_,
0,
if Ic~l = 0 and [c~GI is even,
if Ic~l = 0 and I~GI is odd,
if Ic~l = 1 and I~GI is odd,
if Ic~l --= 1 and I~GI is even,
otherwise.
PROOF. The next map E G + + EG + in the first cofibre sequence is l - g , the difference of the identity map and the multiplication by g map, for some element g of G which depends on 7/. The homology and cohomology long exact sequences associated to the first cofibre sequence have the form
G G + O G + . . . - . t t G G + - Jag G + - J t~ s,7 + - , H~_~ - . ~ _ ~ - ~ . . .
and
"~ ~GT][(~--I"~+~ T T ~ - - I ~ + c~ + G + . . . . • .. - * x t G t . - . H ~ S 7 1 + - + I t c G - . H ~ - .
The Mackey functor t t ~ G + may be identified with the Mackey functor (I t~S°)G defined in Examples 1.1(0. The difference 1 - g may be regarded as a map in B(G). Under the identification of H ~ G + with ( I t ' S ° ) 6 , the first map in the part of the homology long exact sequence displayed above becomes the map from (H~S°)G to (H~S°)G induced by the map 1 - g in B(G). It follows that the cokernel of the map
G 0 ( 1 - g ) , : l i ~ G + -+HGG + is the Mackey functor L(I-Io(S )(e)) defined in Examples 1.1(e). Similar observations reduce the homology and cohomology long exact sequences of the first cofibre sequence to the short exact sequences
o - . L ( H ~ ( S ° ) ( e ) ) --, r I~ s ~ + - . R ( H ~ < ( S ° ) ( e ) ) --, 0
Since I - I G ( s ° ) ( e ) " 0 = HI<(S ;?7), L( I IG(s° ) (e ) ) is zero if lal :/: 0. If lal = 0, then
L(I-I~(S°)(e)) is L(7/) for some act ion of G on 7/. This act ion is the sign act ion of 7//2
on 7/ when p = 2 and a contains an odd number of copies of ~; otherwise, the act ion
is tr ivial . Similar r emarks app ly to L(H~-I (S0) (e ) ) , G 0 R ( t t a _ , ( S )(e)), and
R ( H S ( S ° ) ( e ) ) .
T*Ot S + Notice the frequency with which £t e 7? and I-IaGS~ + vanish. F r o m the
d imension ax iom, we also obta in tha t ~ + t-leG = t t ~ G + = 0 if l a I ¢ 0 . These
vanishing results de te rmine mos t of the homological and cohomological behavior of the m a p s e in our second and the third cofibre sequences.
L E M M A A.2. Let a C RSO(G) .
(a) T h e r n a p e*: ~-~ 0 ~ ~ TJ~(S % t t e S ~ H ~ ( S ) - * * , C ~ J
is mono f o r l a l @ l , 2, epi for Ic~l @0, 1, iso for tc~l @0, t, 2.
a ~ ~ 0 ~ - < S 0 ~ t te(S ) (b) If p = 2, then the m a p e*:,u. G ~ HG(S )
is mono for lal @ 1, epi for Ic~I @ 0, iso for lal @ 0, 1.
The divisibil i ty results involving Euler classes in L e m m a s 4.2, 4.6, and 4.8 of ~ n <0 follow f rom this l emma. Moreover, from this l e m m a and the vanishing ~G~, , for
n E 7/ and n @ O, one can derive all of the zeroes in the first and third quadran t s of o f H * ~0 our s t andard plot ~ e o .
L E M M A A.3. Let a e RSO(G) . Then H~S ° = 0 if lal and la e] are both posit ive or bo th negative.
L e m m a A.2 indicates tha t all of H ~ S ° can be deterrrfined f rom the values of a 0 I tGS for the a in RSO(G) with - 2 _ < ! a t < 2 . If p = 2 , it suffices to know ~ 0 t tGS for
the a in R S O ( G ) with -1 _< Ial _< 1. The next l e m m a describes * 0 H e S on the edges of these two ranges of values for lal.
L E M M A A.4. Let a E R S O ( G ) and let r/ be any nontr ivia l irreducible complex representa t ion of G.
,.~ a + ( 0 a + ( + "~Gu~S° : ker (p : It G S - 4 I t G G ).
Moreover, in all four cases, I-I~(S°)(e) = 0.
P R O O F . Pa r t (d) follows immedia te ly f rom the cohomology long exact sequence associated to the third cofibre sequence. Pa r t (c) follows via dual i ty f rom the homology long exact sequence associated to the third cofibre sequence. For par t (b), consider the d i ag ram
ua+nS0 f TTe+n S + 0 -4 I-I~S ° -4 ~.G .0. G 77
l h
H~+nG+ G
in which the row is f rom the cohomology exact sequence of the second cofibre sequence and the vertical arrow comes f rom the inclusion of an orbit G into St/. Clearly, ~ 0 . . . . S + ttGS = k e r f . By our c o m p u t a t i o n of n G 7/ , the m a p h is mono, so k e r f ~ ker h f . The composi te h f is jus t p. The proof for pa r t (a) is s imilar , but
uses the homology long exact sequence to describe t t - ~ S o as the cokernel of the m a p
I-I~_~ G + -4 ItG_~ S o induced by the collapse m a p G + -4 S °. Dualizing the homology
Mackey functors to cohomology Mackey functors gives the result since the t ransfer is the dual of the collapse map . In all four cases, the group H~(S° ) (e ) is zero either because r (e) is SUljective or because p(e) is injective.
Most of the values of tIGS~ 0 for lal = 0 and la G] @ 0 follow immed ia t e ly f rom the cohomology long exact sequence of the second cofibre sequence and L e m m a s A.1 and A.3.
L E M M A A.5. Let a E RSO(G) with lal = 0 . Then
i ,
u ~ S O R_, ~ G = L,
if laG[ _< 2 and laG] is e v e n ,
if laG[ < - 1 and laG[ is odd,
if laG[_ 2 and laGI is even,
if laGI > 3 and laGI is odd.
114
PROOF. Let r/ be any nontrivial irreducible complex representation. If Ja'GI < 0, then consider the portion
a - r / 0 ,'~ Hc~ S 0 a + ,'~ H G S = c~ r2 i ~ + l S r / i_ia+l--r/¢/0 H G S ~ ~ ' G -~ t l G S r / -~ "~-G =-~'G
of the cohomology long exact sequence of the second cofibre sequence. The left hand term is zero by Lemma A.3 and the right hand term is zero by the same lemma
unless l GI is -1. If -1, then p = 2, a = 4 - 1, I-IGSr ] is R_ by Lemma A.1,
Lia+l--~O and ~G ~, is (:~) by Lemma A.4. The last identification is based on the
observations that ~ must be 2~ and H~S ° is A. By inspection, there are no
nontrivial maps from R_ to {77). Thus, if I GI < 0, the middle arrow must be an
isomorphism.
If JaG1 _> 2, then consider the portion
a + r / - 1 0 , T a + r / - 1 S + r . j ra+r /s r l ~ i ~ a S 0 ~ i a + r / S 0 H G S -+ .LI. G r/ ~ ~-~G =aaG -+ "~G
of the cohomology long exact sequence for the second cofibre sequence. The left and right hand terms in this portion of the sequence nmst be zero by Lemma A.3. Therefore, the middle arrow is an isomorphism.
If p = 2, then the results above reduce the computat ion of I-I~S ° to the
determination of t t~S °, which is A by the dimension axiom, and It~-¢S °, which is
given by the following lemma.
LEMMA A.6. If p = 2, then II~-<S ° ~ R_.
PROOF. Consider the portion
s o G + ° s o
of the cohomology long exact sequence of the third cofibre sequence. By the dimension axiom, the right hand term is zero and the first two terms from the left
are A and AG, respectively. The value of ttG-~S° follows by computat ion.
If p ~ 2 , then we must still determine the value of a 0 HGS when tal = 4-1 or c~ E RSO0(G ). The next three lemmas dispose of the c~ with I~1 = ±1 which are not already covered by Lemma A.3.
LEMMA A.7. Let M be a Mackey functor and f : L - ~ M be a map. If f(e) is a monomorphism, then so is f.
PROOF. The composite f(e) p is a monomorphism and pf(1) = f(e) p.
115
LEMMA A.8. If p 5/= 2, ee e RSO(G), lee, = 1, and leeGI < 0, then I-IGS a 0 = 0.
PROOF. Consider the portion
a--r1 0 ~ a ~ c~ 0 c~ + f 14a+l~ ~.~ ~+i--~7 0 H G S H G S = t t G S r I aa G o H G S ~ H G S -., =
of the cohomology long exact sequence associated to the second cofibre sequence. The left hand term must be zero by Lemma A.3. By Lemma A.1, ~ + I-IGS ~ ~ L. Since
lee+ 1 - ql = 0, I-I~+l- '(S°)(e) is g. The map f: ttGSr/° + + i_iG~+l-nS0 is induced by
the geometric map S ~ + ESr/+ which identifies the points 0 and oo in S ' . From this description, it follows that f(e) is an isomorphism. By the lemma above, f is a monomorphism. Therefore, ~ 0 t tGS must be zero.
LEMMA A.9. Assume that p ¢ 2 , ee e RSO(G), leel = - 1 , and JeeGJ > 0. any nontriviat irreducible complex representation r/,
a 0 ,~, a+rl--1 0 - - a + r l - 1 S + \ I"IGS = coker (tIG S ~ 1 t G r/ ).
of the cohomology long exact sequence for the second cofibre sequence. The right hand term must be zero by Lemma A.3. The first part of the lemma follows immediately. By Lemma A.1, gl G u = R. The map h is induced by the collapse map Sq + + S °. Since J a + r l - 11 = 0,
= ttG ( S , ) ( ) Z.
The map h(e) is an isomorphism by an obvious computation in nonequivariant c~+rl--1 0 ,',., cohomology. If JaGj > 1, then by Lemma A.5, I t G S = L. The only two maps h
from L to R with h(e) an isomorphism have cokernel {g/p}.
If d ~ 0 mod p, then the only maps h: Aid] -* P~ with h(e) an isomorphism
are surjective. Therefore, once we have shown that I t , S ° is A[dz] when
fl E RSO0(G), it will follow from the lemma above that ~a~S° a, G = 0 w h e n l a I = 1 and
lofiJ = 1.
Lemma 4.6 follows from Lemma A.9.
PROOF OF LEMMA 4.6. Let a and fl be elements of RSO(G) with leel = - 1 , G Gt>0, 1~ t=0 , and ]flGj_<0. Let q be a nontrivial irreducible complex
116
representation. Consider the diagram
a + r / - 1 0 LI~S0 R=I - t c S --, , ,G -4 0
1 1 ilia +3+r / - - l~0 a+fl 0
R - - ~ c ~, ~ It G S ~ 0
in which the vertical arrows are given by multiplication by {~ or #~. The rows of ~+~-~ o 1 this diagram are exact by the proof of Lemma A.9. Let y C H G ( S ) ( ) be a
generator and let x 6 H~(S°)(1) be its image. Since p preserves products, p ( f~y ) must be a generator. Thus, @/~y must be a generator and so must @ox. Similarly, p(#~ y) is d/~ times a generator, so #~ y is d~ times a generator. It follows that #~ x is a generator. This proves Lemma 4.6 in the special case where Ic~1---1 and Iofil > 0. The general case follows from the special case and Lemma A.2.
Let a be an element of RSO0(G ). The main difficulty in identifying HGSa 0 with A[d~] is that we must select a representative for a in R0(G) in order to define ~ and d~. To circumvent this difficulty, we work primarily with elements of R0(G) instead of elements of RSO0(G ) in the remainder of our discussion of the additive structure of * 0 t{GS . If c~ is in R0(G), we write H~S ° for the cohomology Maekey functor associated to the image of c~ in RSO(G). To work with elements of I~0(G), we must introduce variants of Definitions 4.5(a) and 4.5(d).
DEFINITION A.10. Observe that the procedure used to produce the element #4 in
Definitions 4.5(a) actually associates a map p: S 2~i ~ S ~¢i to any element ~ ¢ i - ~ i
of R0(G). If a is a nonzero element of R0(G), denote this map, and its image in a 0 I-IG(S )(1), by ~ , Let n0 denote the identity map of S O and 1 E H~(S°)(1). If ¢ is
a nontrivial irreducible complex representation, then let ea,¢ : S 2~i ~ S ¢+~¢i denote
the smash product of the map e : S ° + S ¢ and the map no . We also use %,¢ to
denote the corresponding element in/ t~+*(S°)(1) .
If a and /3 are elements in R0(G) which represent the same element in RS00(G), then n~ and n0 need not be the same class in H~(S°)(1). However, the
class e~,~ in tt~+~(S°)(1) is uniquely determined by the sum a + ¢ in RSO(G). This
uniqueness can be exploited to resolve the problems caused by dependence of ~ on Ct.
LEMMA A.11. Let a and /3 be in R0(G) and let ¢ and r / be nontrivial irreducible complex representations such that a + ~b and /3 + 71 represent the same element in
~+~ S O 1 RS0(G) . Then the cohomology classes %,e~ and e~,~ in H G ( )( ) are equal.
t17
PROOF. We establish the result for three special cases and then argue that the
general case follows from them. Let r/, r/1 , r/2, ¢, ¢1, and ¢2 be nontrivial
irreducible complex representations and let c: S¢1+'2-+ S ¢2+¢1 be the switch map.
Regard a l = ¢ l - r / , a 2 = ¢ 2 - r / , and a = ¢ l + ¢ 2 - 2 r / as elements of R0(G). Let
e : S ° - + S ~ be the usual Eulerclass. The two maps l ^ e a n d e ^ l f r o m S ~ t o S ~+~ are
obviously equivariantly homotopie. On the level of maps,
%2,¢1 = ~ (e ^ 1) and %1,¢2 = c/*~ (1 ^ e).
Therefore, ea2,¢ I and c c~1,¢ 2 are equivariantly homotopic. Thus, e~.2,¢1 and e~1,¢2,
regarded as cohomology classes, are equal. Here, the map c is, of course, absorbed in
the passage to an RSO(G)-grading for tt~S °.
If r/ and ¢1 are equal and e': S o --, S ¢2 is the inclusion, then the trick used
above can also be used to show that 1 ̂ e' : S n --* S ¢1+e2 is equivariantly homotopic to
¢~ 0 1 %2,¢q" Thus, if ct 3 = ¢1 - q51 e I~0(G ), then e' and e~a,¢ 2 are equal in I-IG"(S ) ( ) .
Regard /31 = (¢1 - rh) + (¢2 - r/u) and /32 = (¢1 - r/e) + (¢2 - 711) as elements
of t~0(G ). By three applications of the result just proved for e~2,¢ 1 and %1,¢2' it is
possible to show that e~1,¢ and e/)2, ¢ are equal in HZGI+¢(S°)(1).
If c~ and /3 are in R0(G) and ¢ and r/ are nontrivial irreducible complex representations such that c, + ¢ and /3 + r~ represent the same element in RSO(G), then we can convert the pair (o~, ¢) into the pair (/3, r/) by some combination of the three basic transformations for which the lemma has already been proved. Thus,
%,¢ and eZ,,~ must be equal in H~+¢(S°)(1).
This lemma establishes that the element e z of Definition 4.5(d) does not depend on the choice of c~ and V used in its definition.
LEMMA A.12. If a E RSO0(G ), then ~ 0 ItGS ~A[dc~]. Moreover, if r l is any nontrivial irreducible complex representation, then #a is the unique element of t t~(S°)(1) such that e~ >~ = e~+, and p ( # ~ ) = d~ ~ .
PROOF. Recall the map s: RSO0(G ) --* R0(G) introduced in section 2. Let n
o~ E RSO0(G) and assume that s ( a ) = ~ ¢ i - r / i . Let c% be 0 C R0(G) and, for i = 1
118
k l < k < n , let ak be the element ~ ¢ i - r / i of R0(G ). Denote by d(c%) the integer
i = 1 associated to ak by our homomorphism from l~0(G ) to 7/. For 0 < k < n, let /3 k be the element a k + ¢~+, of RSO(G). We will show by induction on k that
i) Ta%S° is isomorphic to A[d(ak)], a.L G
ii) ~ ~nd ~ ( ~ k ) generate H~k(S°)(1),
iii) t-I~kS ° is isomorphic to (7/), and
iv) e ~ generates H~GkS °.
By the dimension axiom and Lemma A.4, these statements are true for k = 0. Consider the portion
~k-~ + l t~s~k+~ ~ ~k+~ 0 H~kS 0
of the cohomology long exact sequence of the second cofibre sequence. By Lemma A.1, The left hand term is isomorphic to L and the right hand term is zero. By Lemma A.7, the left hand arrow is a monomorphism. Thus, we have a short exact sequence
0 ~ L f l'lC~k+l~0a.a. G o -+ I-I~GkS 0 "+ 0.
Assume that the assertions above hold for some integer k. The element #k+l in a k + l 0 (S)(1) hits the generator e ~ in H~k(S°)(1) by Lemma A.11. Since f(e) is H~ an
isomorphism, we may assume that f(e) takes the generator 1 E 7/= L(e) to the a k + l 0 generator t%+ 1 of H~ (S)(e). It follows that #%+1 and r ( t%+l ) generate
t t ;k+l(s°)(1). Since
P (#%+l )=d(c~k+l ) t%+l and p r ( t % + l ) = p L % + l ,
C~k+l 0 H G S is isomorphic to A[d(ak+l) ]. By Lemma A.4, It@k+lS° is isomorphic to (7/)
= I-IGS is isomorphic to and is generated by e/~+ I. Since Po,~ / ~ and d ( a n ) = do, ~ 0 Aid@
Replacing c%+ 1 by c~, r/k+1 by % and /3 k by a + 7? in the cohomology long exact, sequence above, we obtain the short exact sequence
0 -~ L - . I4~S ° h nO+,S0 ~G -* 0.
Our characterization of #~ in terms of e, #o = h(/~o) and p(#~) follows directly from this sequence.
Two general observations suffice for the proofs of many of the multiplicative
119
relations. Any product involving at least one element in the image of the transfer map v is easily computed using the Frobenius property
xv(y) = r(p(x) y).
Any relation involving an element, like e-m~, obtained by divided some other element by an Euler class may be checked by eliminating the division by the Euler class and checking the resulting relation. The original relation then follows by Lemma A.2.
PROOF OF THEOREM 4.1. We will describe the individual Maekey functors H~S ° of H~S ° by their positions in our standard plot of H~S °. Since
H~(S°)(e) ~Hlal(S°; 7/), it is easy to check that the elements ~1-~ and re_ 1 generate
H~(S°)(e) and satisfy no relations in H~(S°)(e) other than the obvious relation
tl_ ¢ re_ t = p(1). It follows immediately from the structure of the Mackey functors
R_, L, and L_ that the elements r(t~_~), for n > 1, generate the part of tt~(S°)(1) on
the positive horizontal axis. For any positive integer n, p(~,~) = t~_12n. Therefore, ~'~
must generate H2"(;-1)(S%q~ /~ /. The relation r(t~_l) = 2 ~'~ follows from the additive
structure. No other relations involving only ( and t¢-I are permitted by the additive structure. Lemmas A.2 and A.4 ensure that the powers of e generate the part of II~(S°)(1) on the positive vertical axis. These two lemmas also indicate that the
• 0 1 elements e m 4'~, for m, n _> 1, generate the part of HG(S )( ) in the second quadrant.
The same two lemmas indicate that the elements e -m~ and the elements - r n / 2 n + l ~ e r ( t l_ ~ ) generate the parts of H~(S°)(1) on the negative vertical axis and in the
fourth quadrant, respectively. The relations not already verifed follow easily from the additive structure of * 0 I-IGS or from our general observations. The additive structure of H~S ° eliminates the possibility of any unlisted relations involving a single element. Since we have described every possible nonzero product of a pair of generators in terms of the generators, no further relations involving products are possible.
PROOF OF THEOREM 4.9. Again, we describe the individual Mackey functors a 0 * 0 HGS in terms of their positions in our plot of t t G S . Since H~(S° ) ( e )~ Hill(S°; ~),
it is easy to check that the relation ~= ~# = %+# holds for any c~, /3 E RSO(G) with
]od = ]/3] = 0 and that no other relations in H~(S°)(e) hold among the L~. Therefore, for any /3 G RSO(G) with I j31 = 0, L z can be written as a product of the Lc~ included in the proposed list of generators of * 0 t-IGS . The elements LZ, for /3 E RSO(G) with I/3t = o, generate H~(S°)(e) and the elements r(L#), for /3 E RSO(G) with 1/31 = 0 and
[fiG[ > 0, generate the part of H~(S°)(1) on the positive horizontal axis.
Let a and /3 be in RSO0(G ) and let 7 be an element of RSO(G) such that 17t > 0 and 17GI = 0 . The relation pa¢.~ = %+7 follows from Lemma A.11. The relation
120
#o #8 = #~+~ + [(d~d8 - do+/~)/P] r(ta+/~)
follows from ore" characterization in Lemma A.12 of /%+~ as an element, of }t~+5(S°)(1). From this relation, it follows that all of the elements #~ can be constructed from the Iz~ and ~ in our proposed list of generators. By Lemma A.12, the elements #0 and ~ generate all of the u ~ S ° which are plotted at the origin. The
a.~ G
relation #~ e~ = e~+~ indicates that we can construct all the elements e7 from our proposed list of generators. By Lemmas A.2 and A.4, these elements generate all of the o 0 fIGS on the positive vertical axis.
Let ~ E RSO0(G ) and 13,7 • RSO(G) with I~/I =171 = 0 and I/3(31, 1 (31 < 0 . The element ~ro can be obtained from #4 and t~. The relations
P(/J~ ~8) = do to+ 8 = p(do ~ + 8 ) ,
and
follow- from the fact that p is a ring homomorphism. They imply the relations #o ~4 = d ~ + 8 , ~°~8 = 4o+~, and ~p ~ = ~ + ~ since p is a monomorphism in dimensions c~ + fl and fl + 7- These relations indicate that all of the elements ~ can be produced from our proposed list of generators. These elements generate the part of • 0 H(3S on the negative horizontal axis. By Lemmas A.2 and A.4, the elements
T~* S 0 e 6 ~ generate the part of ~G in the second quadrant.
The relations / ~ (e5 i ~ ) = e~ i ~;~+8 and e~ i ~ca = e~ 1*%, for c~ + 7 = fl + ~, may be checked by our general procedure for relations involving division by an Euler class. Together, these relations indicate that our proposed set of generators suffices to construct all of the elements e~ i Ks and therefore to generate the part o f H ~ S ° on the negative vertical axis.
Let fl E RSO0(G) and let cr • RSO(G) with Icrl < 0 and I GI > 0 Recall the class ~o and the virtual representation <c~> from Definitions 4.7. By definition, < ~ + fl> = <c~>, and by the Frobenius relation, ~'<o> r(~o+Z) = 0. Therefore,
# 8 ~'a = ptfl # o - < o > z /<o>
Ptc~+8-<c~>/ ;<0>
b 'o+ ~ .
This relation indicates that our proposed set of generators suffices to produce all of the elements ~'o and therefore the part of H~S ° in the fourth quadrant.
We have now shown that our proposed set of generators does generate I-I~S °. Seven of the relations we have not already established deserve comments . The relation eoep = %+p follows easily from the definition of the guler classes, the Frobenius relation and the product relation for the classes /l~. The relation eO ~ = de_ ~ e~ ~a, for ct + ~ = 7 + ~, follows from the sequence of equations
121
= e-r #~_~ {~
= da_oe~ ~a"
The relations ~ ~, = P ~ + e and ~7 v~ = 0 can be confirmed from the definitions, the Frobenius property, and the relations which have already been established. Given these equations, the relations
e~ (e51 ~) = ~l_~ ~ ,
(G ~ ~)(~ ~ ) -~ = Pe~+v ~;~+~'
and (~71 ~ ) . ~ = 0
follow from our general procedure for checking relations involving classes divided by Euler classes. For the relations eZus = u~+p and ~Zus =d<z>_Z~,~+Z, observe that ~Z can be written as c~¢_<p>~<Z> and that eZ can be written as #vena , for some 7 E RSO0(G ) and some positive integer n. The relations now follow by straightforward computat ions using the definitions, the Frobenius property, and the previously established relations. All of the remaining relations in the theorem follow
* 0 directly from the definitions or the additive structure of H G S . The additive structure of • 0 I-IGS eliminates the possibility of any unlisted relations involving a single element. Since we have described every possible nonzero product of a pair of generators in terms of the generators, no further relations involving products are possible.
122
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JILL]
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[LMM]
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[LIU]
[MAT]
[WIR]
T. tom Dieck and T. Petrie, Geometric modules over the Burnside ring. Inventiones Math. 47 (1978), 273-287.
A. Dress, Contributions to the theory of induced representations. Springer Lecture Notes in Mathematics, vol. 342, 1973, 183-240.
S. Illman, Equivariant singular homology and cohomology I. Memoirs Amer. Math. Soc. vol. 156, 1975.
L. G. Lewis, Jr., The equivariant Hurewicz map. Preprint.
L. G. Lewis, Jr. An introduction to Mackey functors (in preparation).
L. G. Lewis, Jr., J. P. May, and J. E. McClure, Ordinary RO(G)-graded cohomology. Bull. Amer. Math. Soc. 4 (1981), 208-212.
L. G. Lewis, Jr., a. P. May, and M. Steinberger (with contributions by a. E. McClure). Equivariant stable homotopy theory. Springer Lecture Notes in Mathematics, vol. 1213, 1986.
H. Lindner, A remark on Mackey functors. Manuscripta Math. 18 (1976), 273-278.
A. Liutevicius, Characters do not lie. Transformation Groups. London Math. Soc. Lecture Notes Series, vol. 26, 1976, 139-146.
T. Matumoto, On G-CW complexes and a theorem of J. H. C. Whitehead. J. Fac. Sci. Univ. Tokyo 18 (1971/72), 363-374.
K. Wirthmiiller, Equivariant homology and duality. Manuscripta Math. 11 (1974), 373-390.
THE EQUIVARIANT DEGREE
by
Wolfgang L~ck
O. Introduction
Abstract. In this paper we study the possible values deg fH,
H c G for a G-map f : M ~ N if M and N are compact smooth G-
manifolds and G a compact Lie group. We generalize results about
maps between spheres of G-representations. We give applications
to one-fixed point actions and G-surgery. We prove that the un-
stable H-homotopy type of the sphere of the H-normal slice
SQ (MH,M)x for x 6 M H is a G-homotopy invariant of M.
Survey. As an illustration we state a consequence of our main re-
sult in a very special situation where it is easy to formulate.
Let G be finite. Consider a compact smooth G-manifold M such that
M H is non-empty, connected and orientable for all H c G. Assume
either that G is nilpotent or that dim M H ~ dim M K -2 holds for
H c K, H, K ~ Iso(M) = {G x I x £ M} Here and elsewhere G denotes " x
the isotropy group {g 6 G I gx = x} of x 6 M. The set of finite G-
sets S with Iso(S) c Iso(M) is an abelian semi-group under disjoint
union. Let A(G,Iso(M)) be its Grothendieck group. The cartesian
product induces the structure of a commutative ring with unit on it.
Let Con(G) be the set of conjugacy classes of subgroups of G and
C(G) be the ring ~ ~. Then A(G,Iso(M)) is a subring of C(G) Con(G)
by identifying S with (card sH I (H) £ Con(G)). For a G-selfmap
f : M ~ M define DEG(f) 6 C(G) by (deg fH I (H) £ Con(G)).
Theorem A.
a) DEG(f) £ A(G,Iso(M) c C(G).
b) If H c G is a p-group then:
deg f ~ deg fH mod p.
t24
c) If G has odd order and deg fH 6 [+1} for each H c G, then we
have for all H c G:
deg f = deg fH. o
This theorem is well known for M as the one-point compactification
V c of a G-representation V. The proof for V c uses the equivariant
Lefschetz index and Smith theory. These methods do not suffice
for M a G-manifold. Our main tool is quasi-transversality and the
notion of a local degree.
The notion of the degree is used to classify G-hcmotopy classes
of G-maps f : V c ~ W c (see tom Dieck [6'], p. 213, Laitinen [14],
Tornehave [21]) and G-homotopy types of G-homotopy representations
(see tom Dieck-Petrie [8]). It plays also a role in equivariant
surgery theory (see for example Dovermann-Petrie [11], LHck-
Madsen [17]). We give a survey over the various sections.
In section one we define the fibre transport tPM of the tangent
bundle of a G-manifold and the notion of an O(G)-transformation
: f tp N ~ tPM for a G-map f : M ~ N. Roughly speaking,
assigns to each point x in M a G-map (not necessarily a G -homo- x
topy equivalence) TN~x ~ TM~ such that certain compatibility
conditions hold. Using ~ we get a one-to-one-correspondence between
local orientations of M H at x and N H at fx for each H c G and
x 6 M H. This enables us to define the equivariant degree
DEG(f,~) 6 C(N) = ~ ~ ~ in section two. (H) E Con(G) , (N H)/WH
o
In section three the Burnsidering A(G,~ ) of a compact Lie group
with respect to a family ~ is treated. We identify [vC,vC] G and
125
A(G,Iso(V)) for a G-representation V. We introduce in section
four a multiplicative submonoid Endtp N c C(N) and prove
DEG(f,~0) 6 Endtp N for any f and ~0 in section five. We will see
that Endtp N does not involve f and ~ but depends only on the compo-
nent structure of N. The main idea of the proof is best explained
in the special case where G is finite and all N H are non-empty
and connected. Then C(N) = C(G) = ~ ZZ and Endtp N is A(G,Iso(N)). Con (G)
Choose y in N G and make f quasi-transverse to y. Then f-1 (y) is
finite and for each x 6 f-1 (y) f looks like a (not necessarily
linear) norm-preserving Gx-ma p TxM ~ TyN in a Gx-neighbourhood
of x. Consider a G-orbit c of f-1 (y). For each x in c we obtain
Gx-maps TM c ~ TNy by f and TNy ~ TM c by ~. Their composition
TNy ~ TNy defines an element in A(Gx,ISo(TNy)). Its image in
A(G,Iso(N)) under the induction homomorphism for G x c G is inde-
pendent of the choice of x and denoted by d(c). Let d be the sum
Zd(c) running over c 6 f-1 (y)/G. Since the global degree can be
computed by local degrees, DEG(f,~0) 6 C(G) is just d 6 A(G,Iso(N)).
Roughly speaking, we have counted the local degrees orbitwise in
the Burnside ring to get the global degree.
Section six contains some examples to illustrate our results. We
give an elementary proof of the following known statement (see
Atiyah-Bott [I], Browder [4], Ewing-Stong [12]).
Corollary B. There is no closed G-manifold M with dim M ~ I such
that each M H is connected and orientable and M G a single point
if G is the product of a p-group and a torus, m
126
It is of special interest to choose ~ : f tPN ~ tPM as an
O(G)-equivalence i. e. all TN~x ~ TM~ are Gx-homotopy equi-
valences. Then another choice of ~ would change the equivariant
degree only by a unit. Moreover, we have:
Theorem C. A normal G-ma~ f : M ~ N can be change d into a G-
homotopy equivalence by equivariant surgery only if there is an
O(G)-equivalence ~ : tPN ~ f tPM with DEG(f,~) = I. o
The existence of an O(G)-equivalence ~ is related to the notion
of the equivariant first Stiefel Whitney class w M of a G-mani-
fold. In section seven we relate tPM und w M and show that the
existence of an O(G)-equivalence ~ : f tPN ~ tPM is equivalent
to f w N = w M. We prove:
Theorem D. I_ff f : M ~ N is a G-homotopy equivalence w~e have
f w N = w M. o
This implies the unstable version of the stable result in Kawakubo
[13].
Corollary E. I_~f f : M ~ N is a G-homotopy equivalence, w_~e @et
for x 6 M:
TMC ~G TN~x " [] x
Our setting and proofs would be much simpler if we supposed that
all fixed point sets are non-empty, connected and orientable. Un-
fortunately, such conditions are unrealistic in the study of G-
manifolds. Hence we make no assumptions about the existence of
127
G-fixed points or about the connectivity or orientability of the
fixed p o i n t s e t s a n d do n o t d e m a n d ~ ( fH) b e i n g b i j e c t i v e . o
Our notion of the equivariant degree using O(G)-transformations
has some advantages compared with the one using fundamental
classes. It is in this generality much easier to state elemen-
tary properties like bordism invariance or the computation by
local degrees in our language. We have the global choice of
instead of the various choices of fundamental classes [M H] and
[NH]. Notice that the choice of [M H] is independent of the one
of [M K] for (K) • (H) and [NK]. Hence in the case of fundamental
classes the interaction between the various fixed point sets are
not taken into account, what is done in our setting. It seems to
be difficult, or even impossible, to state some of our results
by means of fundamental classes. For example, the statement of
example 6.5 makes no sense if it is formulated with fundamental
classes and in example 6.3 there must appear signs because we
can substitute [M H] by -[M H] and thus change the corresponding
degree by a sign. The advantages of our approach for the notion
of an equivariant normal map is worked out in L~ck-Madsen [17].
(see also theorem C above and example 2.8).
Conventions: We denote by G a compact Lie group unless it ex-
plicitly is stated differently. Subgroups are assumed to be
closed. A G-representation is always real, A G-manifold M is a
compact smooth G-manifold with smooth G-action and possibly
non-empty boundary. We call a component C of M H an isotropy
component if there is a x in C with isotropy group G x = H. We
say that M fullfills condition (~) if it satisfies the conditions
128
i) and ii) or the conditions i) and iii) below.
i) C # {point} for all C 6 ~ (MH), H c G. o
i i ) I f C £ ~ (M H) i s an i s o t r o p y c o m p o n e n t , C >H i s o
H {x £ C 1G x ¢ H} a n d H c G we h a v e d i m C > H + 2 ~ d i m C
iii) G is finite and nilpotent.
A G-map f : M ~ N respects always the boundary and we assume
6 Mo(MH) , D £ Mo(NH), H c G with fH(c) c D. dim C dim D for all C
Acknowledgement. The author wishes to thank the topologists at
Arhus for their hospitality and support during 1985 - 1986 when
the main part of this paper was written. The author is indebted
to Ib Madsen and Erkki Laitinen for their useful comments.
I. The fibr e transport.
We organize the book-keeping of the components of the varbus fixed
point sets and their fundamental groups for a G-space as follows.
We recall that an object of the fundamental groupoid ~ (Y) of a
space Y is a point in y and a morphism Yo ~ Yl is a homotopy
class of paths from Yl to Yo" The orbit category O(G) has the ho-
mogenous spaces G/H as objects and G-maps as morphisms.
Definition 1.1. The fundamental O(G)-groupoid uGx of a G-space X
is the contravariant functor uGx : O(G) ~ {groupoids} sending
G/H to U(X H) = H(map(G/H,x)G).
In general an O(G)-category resp. O(G)-groupoid is a contravariant
129
functor from O(G) into the category of small categories resp.
groupoids. We recall that a groupoid is a category whose mor-
phisms are all isomorphisms. An O(G)-functor F : C ~ D between
O(G)-categories is a natural transformation. Let I be the cate-
gory of two objects O and I and three morphisms ID : O ~ O,
ID : 1 ~ I and u : O ~ I. We define an O(G)-transformation
: F ° ~ F I between O(G)-functors F ° and F I : C ~ D as an
O(G)-functor ~ : C x I ~ D with C I i = F i. Given a second O(G)-
transformation 4: F 1 ~ F2, let the composition ~ ~ ~ : F o ~ F 2
be determined by ~ ~(id,u) = ~(id,u) o ~(id,u) : (x,O) ~ (x,1)
for all x 6 C. One should think of an O(G)-functor F : C ~ D as
a collection of functors F(G/H) : C(G/H) ~ D(G/H) and of an
O(G)-transformation ~ : F o ~ F I as a collection of natural trans-
surgery transfer maps (C,a,U) " :Lm(R) JLm+n(S) to the
intermediate L-groups, and show that they are
compatible with the Rothenberg exact sequences.
An involution R ~R;r ,r on a ring R determines a
duality involution *:P(R) ,~(R);P ,P =HOmR(P,R) on
the additive category ~(R) of f.g. projective R-modules
by
R X P , e ; (r,f) , (x , f(x).r) ,
e(P) : P ~ P ; x , (f ~ f(x))
The duality involution on P(R) determines involutions
on the algebraic K-groups
* : K 0 ( R ) ' K 0 ( R ) ; [ P I ' [ P ] ,
* : K1 (R) , K1 (R) ;
T(f :P 'Q) , T(f :Q 'P )
and also on the reduced K-groups
Ki(R) = coker(Ki(Z) ,K.(R)) (i--0 I)
The intermediate quadratic L-groups L~(R) of a
ring with involution R are defined for *-invariant
subgroups X~Ki(R) (i=0, I), such that x EX for all xEX.
The intermediate L-groups for X=<0>,Ki(R) are written
as
K0 ( R ) ( 0 ) ~ K I ( R ) s L . ( R ) = L ~ ( R ) , L . ( R ) = L . ( R )
( 0 } ~ K 0 ( R ) K I ( R ) I . , ( R ) = L , ( R ) = L ~ ( R ) = L , ( R )
232
For *-invarJant subgroups XC_X'C_Ki(R) there is defined a
Rothenberg exact sequence
with
, L X ( R ) ~ LX' ( R ) n n
, ,,, H n ( z 2 ; X ' / X )
............ : L X (R) n-I
) o , °
Hn(z2 ;X'/X ) :
{aEX' /Xla =(-)na}/(b+(-)nb IBEX' /X}
See RanickJ [ ] 3] , [ ]4] for further details.
We consider first the torsion case XC_K] (R).
A representation (C,U) of R in D(S) determines a
transfer map in the abso]ute torsion groups
(C,U) :KI(R) 'KI(S) (Example 1.8), and also in the
' K (R) (S) By reduced torsion groups (C,U)' : 1 'KI "
definition, D(S) is the homotopy category of finite
chain complexes of based f.g. free S-modules. We sha] 1
now make use of the bases.
Proposition 9.1 Let (C,(I,U) be a symmetric
representation of R in Dn(S) , for some rings with
involution R,S.
i) For any *-invariant subgroups XC_K 1 (R) , YC_K 1 (S) such
:C n-* that (C,U) "(X)C_Y and T(~:C )6Y there are defined
transfer maps in the intermediate torsion L-groups
' LX(R) : L Y (S) (n>0) (C,~,U)" : m m+n
ii) For any *-invariant subgroups X~X'~KI (R) ,
Y~Y'GKI(S) such that (C,U) !(X)~Y, (C,U) !(X')~Y ' , T(~)6Y
there is defined a morphism of Rothenberg exact
sequences
233
LX, ^ Z2 ' LX(R) , (R) ......... ~ Hm( ;X'/X) ~ ... m m
( c , a , u ) " ( c , ~ , u ) " ( c , u ) '
Y ' ~m+n ' LYm+n(S) --~ Lm+ n (S) --~ (~o ;Y'/Y)~ ~ ....
Proof : The transfer map in the reduced torsion groups
(C,U)" :KI (R) ~KI (S) is such that
*(C,U) ! = (-)n(c,u)!* : K] (R)
Let m=2i. For any nonsingular (-)i-quadratic form (M,~)
on a based f.g. free R-module M=R k the n-dimensional p
(-)i-quadratic Poincare complex (~C,@) representing k
! (C,U) "(M,%b) has reduced torsion
T ( ( I + T ) e o : ~ g C n - ,@C) k k
' i * * -- (C,U)' T(~+(-) ~ :S :M ) 6 K1 (S)
image of T(~+(-) i@*)6KI(R).~ Similarly for the m=2i+l
and formations.
[]
Next, we consider the projective case X~Ko(R) . It
is more convenient to work with the preimage of X in
Ko(R) , so we regard X as a *-invariant subgroup of
Ko(R) such that [R]EX.
Given a ring S let E(S)=D(~(S)), the homotopy
category of finite-dimensional f.g. projective S-module
chain complexes. A representation (C,U) of a ring R in
E(S) determines transfer maps in the algebraic
K-groups
! (C,U)" : Ki(R ) = Ki(•(R))
234
Ki(S ) = Ki(?(S)) (i=O, I)
(Example 1.8). For n~0 let En(S)=Dn(D(S)), the full
subcategory of E(S) with objects n-dimensional f.g.
projective S-module chain complexes. An involution on S
determines the n-duality involution C :C n- on ~n(S).
Proposition 9.2 Let (C,~,U) be a symmetric
representation of R in ~n(S), for some r~ngs with
involution R,S.
i) For any *-invariant subgroups X~K0(R) , Y~K0(S) such !
that [R]6X, [S]6Y, (C,U)(X)~Y~K0(S) there are defined
transfer maps in the intermediate class L-groups
' LX(R) , L Y (S) (n~O) (C,~,U)" : m m+n
ii) For any *-invariant subgroups X~X'~K0(R) , !
Y~Y'~K0(S) such that [R]EX, [S]EY, (C,U) " (X)~Y,
(C,U) " (X')~Y' there is defined a morphism of Rothenberg
exact sequences
, LX(R) , LX' (R) m m
, H m ( z 2 ; X ' / X )
I I l (c,a,u) (c,~,u) (
!
C,U) '
y, ^ + -- L +n(S) ) Lm+n(S) ---* I! m n(Z2;Y'/Y) I . o , °
[]
The proof of 9.2 is somewhat more involved than
that of 9.1.
A splitting (B,r,i) in ~ of an object (A,p) in the
idempotent completion ~ is an object B in ~ together
with morphisms r:A ~B, i:B ~A in ~ such that
ri = 1 : B ......... ~ B , Jr = p : A , A
Lemma 9.3 A functor of additive categories F:~ ~
235
extends to a functor F: ,~ if and only if for each
object (A,p) in ~ the object (F(A),F(p)) in B has a
splitting in ~. Any two such extensions of F are
naturally equivalent.
Proof: It is clear that the splitting condition is
necessary for F to extend to {, so we need only prove
that it is sufficient. For each object (A,p) in
choose a splitting (B,r,i) of the object (F(A),F(p)) in ^ ^
~, and set F(A,p)=B, with (B,r,i)=(F(A), I, ]) for
p=l :A JA. For a morphism f : (A,p) ~(A' ,p' ) let
i f r' ~(f) : {(A) = B , A , A' , {(A') = B'
[]
An additive category A is idempotent complete if
the functor ~ ,~;A ,(A, I) is an equivalence of
categories. Applying 9.3 to I :A 'A we have that A is
idempotent complete if and only if every object (A,p)
in ~ splits in A. If ~ is idempotent complete every
functor F:~ :~ extends to a functor F:A ........ ,~, namely
the composite of and an equivalence S
For any ring S the additive category ~(S) of f.g.
projective S-modules is idempotent complete, with every
object (A,p) in ~(S) split by the triple (B,r,i)
defined by
r : A ~ B = im(p) ; x , p(x) ,
i = inclusion : B , A
This is the special case n=0 of:
Lemma 9.4 For any ring S and any n~O the homotopy
category En(S ) of n-dimensional f.g. projective
S-module chain complexes is idempotent comp]ete. 2
Proof: For every chain homotopy projection p=p :D ~D
of an object D in En(S) there exists by Lemma 3.4 of
LUck [7] an (n+l)-dimensional infinitely generated
0 ~
. :~
~
. !~
rT
~ 0
~
rt
rt
0 ~
~
v ~
R
~ D
0
v f/
)
~'1
t't
0 ~
, k~
~ 0
'~
0 II
',,ID
I ,~
-
v v
+ r~
v
Cl
f:I
oo
t ~
v II v v
rt
t~
0 0 r,t
°° I
v • .
~
=1,
~I
v
~ o
~
+
~ ~
v
~ 0
v ~
N
0
0 ~
rT
0 v
< 0
0 ~,
0 lf
f' ~
' ~
. 0
~
~u
0 ~
~ ~
~h
.
0 0
X- 1 >
v
~e
v
0 < 0
='0
~.,
0
~ 0
rr
f3
N
rt
0
ffl
0 <
rt
0
0 ~
[]
0 0 ~
"
0 ~
" ~
" I,
~
0 ..
0 "~
"~
/ 1
0
'~
0 ".-
4 G
' ~
"
n I~
'
0 ~
r
0"
G
~ v
m
~ v
°.
0 I 0 D
:r
0 0
0 r-r
'~
0
mr
237
F = -@(C,~,U) : B(R) ' ~[n(S) ; R ) C
[]
The proof of 9.2 is now completed by observing
that the transfer map in the projective class groups !
( C , U ) " : K o ( R ) ,K0(S) is such that
' )n ' *(C,U)" = (- (C,U)" * : K0(R) : K0(S)
Remark 9.6 Our methods also apply to construct
algebraic surgery transfer maps in the round L-groups
L~X(R) of Hambleton, Ranicki and Taylor [4], which are
defined for *-invariant subgroups X~KI(R). For any
symmetric representation (C,~,U) of R in En(S) and any
*-invariant subgroup X~KI (R), Y~KI (S) such that t
(C,U) "(X)~Y there are defined round L-theory transfer
maps
' LrX(R) , L rY (S) (m~0) (C,~,U)" : m m+n
which are compatible with the round L-theory Rothenberg
exact sequences.
D
Remark 9.7 The connection established in %8 between the
algebraic and geometric surgery transfer maps extends
to the intermediate cases, and also to round L-theory.
[]
Remark 9.8 Our algebraic constructions apply also to
the E-quadratic L-groups L,(R,E), which are defined for
a ring with involution R and a central unit £ER such
that EE=I. L2i(R,E ) (resp. L2i+l(R,e)) is the Witt
group of nonsingular (-)iE-quadratic forms (resp.
formations) over R. A symmetric representation (C,~,U)
of R in Dn(S) such that U(E)=~:C ....... ,C for a central
unit
238
~6S with ~=I induces transfer maps
!
(C,a,U) " : L m ( R , ¢ ) , Lm+n(S,~ ) (m~O)
Hitherto we considered the
L,(R, I)=L,(R) , with 0=I£S.
c a s e c = 1 6 R f o r which
[3
Appendix l . Fibred intersections
! ! The proof of Pgeo=P~Ig_
algebraic properties of the
bundle F ,E P :B with the
n-dimensional manifold it is
the algebra:ic and geometric
coincide more directly, using
theory of Hatcher a n d Quinn
in %8 makes heavy use of
the L-groups. For a fibre
fibre F a compact
possible to verify that
surgery transfer maps
the bordism intersection
[6] to obtain fibred
versions of the geometric intersection forms (resp.
formations) used by Wall [22] to define the surgery
obstruction of a highly-connected even (resp. odd-)
dimensional normal map. The quadratic kernel of the
pullback normal map is the fibred intersection form
(resp. formation) both algebraically and geometrically.
We now sketch the argument for the intersection pairing
in the even-dimensional case, leaving the
sel f-intersection function ~ and the odd-dimensional
case to the interested reader.
Given two maps vi :Qi )M (i=l ,2) let E(Vl ,v 2 ) be
the pointed space of triples (Xl ,x2,w) defined by
points xiEQi and a path ~:[0,[] )M from ~(0)=Vl(X 1 ) to
~(1)=v2(x2) , so that there is defined a homotopy fibre
square
E il'v2) ' i 1 t vl
v 2 Q 2 ...... ~ M
239
Given a stable vector bundle ~ over a space M let fr
~n (M,~) be the bordism group of n-manifolds N equipped
with a map N ~M and a compatible stable bundle map
~N ~" For trivial ~ this is the usual framed fr S
cobordism group Qn (M)=~n(MV{*})" For
vl =v2 :Q] =Q2={ *} ,M the homotopy pullback is the loop
space, E(* *)=QM
Now suppose that M is an m-manifold, and that
vi:Q i ,M is an immersion of a qi-manifold Qi (i=1,2)
such that vl (QI) intersects v2(Q2) in general position.
Let QI~Q2 denote the corresponding
(ql+q2-m)-dimensional submanifold of M. The bordism
Jnvariant of the intersection ([6,2. I ]) is the bordism
class
)%(Vl,V2) = [QI~Q2]
60 fr m(E(v v ) ~Q2~TM) ql+q2 - 1' 2 '~QI
If Q1 and Q2 are (q]+q2-m+I)-connected the map
E(*,*)=QM JE(Vl ,v 2) induces an isomorphism ([6,3. I ])
fr (QM) O~rl+q2 -m(E(*'*)) = Oql+q2_ m
fr _m(E(v 1 v2 ) ~Q2~TM) Oql+q2 ' '~QI
which is used as an identification.
Let (f,b) :M ~X be an (i-1)-connected
2J-dimensional normal map with a ~l-isomorphism
reference map X ~B, with the surgery obstruction
~,(f,b)=(Ki(M),k,~)6L2i(Z[~|(B) ]) defined as in Chapter
5 of Wall [22]. Let Vl,V2, ... ,Vk be a base of the
kernel f.g. free Z[~I(B) ]-module Ki(M)=~i+l(f).
Represent each vj6Ki(M ) by a pointed framed immersion i v :S ~M with a nullhomotopy in X. The values taken by
3 i the (-) -symmetric form (Ki(M),X) on the base elements
are just the bordism intersections
240
k(vj,vj,) f r 6 n 0 (E(vj,vj,),p i~ i~fM)
S S
fr = n O ( Q M ) = H o ( Q M ) = Z [ K [ ( B ) ]
(l~j, j'~k)
Now let (g,c) :N ,Y be the (i-l)-connected
(n+2i)-dimensional normal map with a ~l-isomorphism
reference map Y ,E obtained from (f,b) :M ",X by the
pullback of the fibre bundle F ,E ~ ,B along X :B. The • S i pointed framed immersions v : ~M (I~j~k) with 3
nullhomotopies in X lift to pointed framed immersions i w .:S XF ,N with nullhomotopies in Y. On the chain
3 level this corresponds to lJ fting the kernel
i i Z[~I(B) I-module chain complex C(f" )=S Ki (M)=~S Z[KI (B) ]
k
to the kernel Z[K1 (E) ] - m o d u l e chain complex
' ic( C(g')=~)S F~). The bordism intersections k
fr(E(wj,w3, ),~ ~ ) k(wj,wj, ) E n n SIXFe~S XF~TN
fr = n n (QMXF,~F) (l~j, j'~k)
are the images of the bordism intersections )~(vj,vj,)
under the geometric bordism transfer map
fr p" = -XF : •0 (riM)
f r Qn ( QMXF, ~F ) ;
X J XXF
The Poincare duality isomorphism of based f.g. free
Z[~I(B) ]-modules
(k(vj,vj,)) :
* t si ' 2i-* SiK (M) ~ C(f' ) (M C(f') = - - = K ) 1 i
i
is lifted to the Poincare duality chain equivalence of
241
chain complexes of based f.g. free Z[KI(E) l-modules
' n+2i-* (X(wj,wj,)) : C(g') = ¢siC(F) n-* k
!
", C ( g ' ) : e s i c ( ~ ) k
Using the Poincare duality Z[~I(E) l-module chain
equivalence [F]~-:C(F) n-* ~C(F), the action of ~M on
the ~] (E)-equivariant homotopy type of F and Hurewicz
maps there is defined a commutative diagram
flfr(f~M ) ~ , H0(QM) ~- , Z[~I(B)]
I l Iu
I [ F I ~ -
f r ( n M X F :o F ) ~ It ( n M X F ) ~ H n ( C ( F " ) ® Z [ ~ I ( E ) l C ( F O ) ) • •n ' n
The anticlockwise composition gives the geometric
surgery transfer Pgeo on the level of intersections,
while the clockwise composition gives the algebraic
surgery transfer P a l g
Appendix 2. A counterexample in symmetric L-theory
An n-dimensional Poincare fibration F .... ~E P ,B does
not in general induce transfer maps in the symmetric !
l,-groups p" : L m ( Z [ K I ( B ) ] ) ~Lm+n(z[~I(E) ]), either
algebraically or geometrically. It is not possible to
define p geometrically since the symmetric L-groups
are not geometrically realizable (Ranicki [ 16,7.6.8]).
There are two obstructions to an algebraic definition !
of p , which requi~es the lifting of an m-dimensional
symmetric Poincare complex (C,~) over Z[~I(B) ]
representing an element (C,#)6Lm(Z[KI(B) ]) I
(m+n)-dimensional symmetric Poincare complex
over, Z[KI(E),] representing the putative
p" (C,#)=(C" ,#" )ELm+n(z[KI(E) ]). The symmetric
are not so it cannot be assumed 4-periodic,
to an ! !
( c " , ~ ' )
transfer
L-groups
that (C,#)
242
is highly-connected as in the quadratic case. In the
following discussion we assume that the fibre F is
flnJ te, and that the chain complex C consists of based
f.g. free ~[~i (B) ]-modules. The two obstructions to ! !
] ifting (C,#) to ( C ' , ~ " ) are given by:
i) it may not be possible to lift C to a based !
f.g. free Z[K l (E) ]-module chain complex C" with a ! ! ! !
filtration FoC'C_FIC'C...C_FmC" =C" such that the
connecting chain maps between successive filtration
quotients are given up to chain homotopy by
' ' r #(Or) = p#(d C) : FrC'/Fr_IC" = S p
! !
S(Fr_IC" /Fr_2C' = srp#(Cr_l ) (l~r~m)
r # where S denotes the r-fold dimension shift and p
the functor of %1
is
# p = -®(C(F),I:) : 3(Z[~I(B> 1) , Dn(Z[~I(E)]) ,
!
ii) even if C' exists, it may not be possible to ¢
lift the m-dimensional symmetric Poincare structure # ¢
on C to an (m+n)-dimensional symmetric Poincare ! !
structure #' on C" .
If C can be assembled over B in the sense of Ranicki !
and Weiss [2(3] then it can be lifted to C ° , but in
general it is not possible to assemble Z[~I(B) I-module
chain complexes, so already i) presents a non-trivial
obstruction to the existence of transfer in symmetric
L-theory. Even if the obstruction of i) vanishes (e.g.
if B is an Eilenberg-MacLane space K(~I(B),I)) then ii)
may present a non-trivial obstruction. This is
illustrated by the following example, which exhibits
the failure of a projection of rings with involution
p:S ,R=S/(I-t) (t = central unit E S, ~=t-IEs) to 1
induce an S -bundle symmetric L-theory transfer map
' O(R p" :L ) ........... ,LI(S) analogous to the SI-bundle quadratic !
L-theory transfer map p" :L0(R ) ~L[(S) (cf. 4.7). The
243
! ! ! transfer p" (C,¢)=(C" ,¢" ) of a 0-dimensional symmetric
Poincare complex (= nonsingular symmetric form) (C,#) k
over R with C0=R is defined if the symmetric kXk
matrix
#0 = (@0) 6 Mk(R)
!
can be lifted to a kXk matrix ~06Mk(S) !
p(#~)=#0~Mk(R) and
t ! * !
t#; - (#6) = ( Z-t)# i 6 Mk(S)
such that
! t * for some symmetric kXk matrix #i=(#i) EMk(S) , so that
! ! •
( C ' , ¢ ' ) : i s a 1 - d i m e n s i o n a l s y m m e t r i c P o i n c a r e c o m p l e x t
over S w:ith C'=C(l-t :S k :sk). In particular, for
2 2 S = Z2 [Z2XZ2 ] = Z2 [t,u]/(t -l,u -I)
= t , u = t+u+l ,
p : S ' R = Z2[Z2 I = Z2[u]/(u2-1) ;
t ~ ] ~ U J U
the transfer is not defined for the 0-dimensional J
symmetric Poincare complex (C,#)=(R,u) over R, for !
although C can be lifted to C" and #0 can be lifted to ! !
#0 there does not ex:ist a symmetric #~. Both the
obstruct:ions to i) and ii) vanish for the visible
symmetric L-groups VL (Z[~]) of Weiss [23] provided
that B is an Eilenberg-MacLane space K(~I (B), I) , in
which case there are defined transfer maps
! L m VL m+n p : v ( Z [ ~ I ( B ) 1) , ( Z [ ~ I ( E ) I ) .
REFERENCES
[ i ] W.Browder and F.Quinn
surgery theory for G-manifolds and
stratified sets
Proceedings 1973 Tokyo Conference on
244
Man~ folds, Tokyo Univ. Press, 27-36 (1974)
[2] D.Gottlieb S
Poincare duality and fibrations
Proc. A.M.S. 76, 148-150 (1979)
[3] I.Hambleton, J.Milgram, L.Taylor and B.Williams
Surgery with finite fundamental group
Proc. Lond. Math. Soc. (3) 56, 349-379 (1988)
[4] l.Hambleton, A.Ranicki and L.Taylor
Round L-theory
J. Pure and Appl. Alg. 47, 131-154 (1987)
[5] I.Hambleton, L.Taylor and B.Williams
Maps between surgery obstruction groups
Proc. 1982 Arhus Topology Conf.,
Springer Lecture Notes 1051, 149-227 (1984)
[6] A.Hatcher and F.Quinn
Bordism invariants of intersections of
submanifolds
Trans. A.M.S. 200, 326-344 (1974)
[7] W.L~ck
The transfer maps induced in the
algebraic K 0- and Kl-~r0ups by ~ fibration I.
Math. Scand. 59, 93-121 (1986)
[B] W.L~ck and I.Madsen
Equivariant L-theory II.
to appear
[9] W.L;Jck and A.Ranicki
Chain homotop~ prooections
to appear in J. of Algebra
[I0] H.Munkholm and E.Pedersen
The sl-transfer i___nn surgery theory
Trans. A.M.S. 280, 277-302 (1983)
245
[ 1 1 ] F.Quinn
geometric formulation of surgery
Princeton Ph.D.thesis (1969)
[121 S
Surgery on Poincare and normal spaces
Bull. A.M.S. 78, 262-267 (1972)
[ 1 3 ] A.Ranicki
Algebraic L-theory I__~. Foundations
Proc. Lond. Math. Soc. (3) 27, 101-125 (1973)
[14] The algebraic theory of surgery I. Foundations
Proc. Lond. Math. Soc. (3) 40, 87-192 (1980)
[15] The algebraic theory of surgery II.
Applications to topology
Proc. Lond. Math. Soc. (3) 40, 193-283 (1980)
[161 Exact sequences i__nn the algebraic theory o f surgery
Mathematical Notes 26, Princeton (1981)
[171 The algebraic theory of finiteness obstruction
Math. Scand. 57, 105-126 (1985)
[181 The algebraic theory of torsion I. Foundations
Algebraic and Geometric Topology,
Springer Lecture Notes 1126, 199-237 (1985)
[191 Additive L-theory
Mathematica Gottingensis 12 (1988)
[20] A.Ranicki and M.Weiss
Chain complexes and assembly
Mathematica Gottingensis 28 (1987)
[21] C.T.C.Wall d
Poincare complexes
Ann. of Maths. 86, 213-245 (1970)
246
[ 2 2 ] Surgery on compact manifolds
Academic Press (1970)
[23] M.Weiss
On the definition
preprint
of the symmetric
[24] G.W.Whitehead
Elements of homotopy
Springer (1978)
t h eory
W.L~ck: Mathematisches Institut,
Georg-August Universit~t,
Bunsenstr. 3-5,
34 G~ttingen,
Bundesrepublik Deutschland.
A.Ranicki : Mathematics Department,
Edinburgh University,
Edinburgh EH9 3JZ,
Scotland, UK.
SOME REMARKS ON THE KIRBY-SIEBENMANN CLASS R. J. Milgram
In this note we study the relations that hold between the Kirby-Siebenmann class { K S } • H4(BsToP; Z/2) and the first Pontrajagin class.
The first result is that that the natural map p0 : BSTOP ~ B s e does not detect { K S } no mat ter what coefficients might be used. However, the homology dual of { K S } is in the image of the Hurewicz map
lr4(BsToP) ~ H4(BsToP; Z/2).
In fact there is a unique non-zero element [KS] • z d B s T o P ) of order 2, and po([KS]) # 0 • 7q(Bsa) . In particular this implies that w4 + { K S } is a mod(24) fiber-homotopy invariant of SPIN-TOP bundles. However, it is interesting to ask what happens when w2 is non-zero. To understand this we introduce an intermediate classifying space, BTSG for which we have a factorization
f P~ po = p" f , BSTOP ' BTSG BSG.
BTSG is univeral for the vanishing of transversality obstructions through dimension 5. Additionally, BTSa is built out of finite groups (Z/2-extensions of the symmetric groups S,~) in the same way that BSG is constructed from the S , . As a result, explicit construction of homotopy classes of maps into BTSG is often possible.
We show that H4(BTSG; Z/2) = Z /2 (~ Z/48 and that the homology dual of the Kirby-Siebenmann class maps to 24 times the second generator. Thus, this transversality theory does detect {KS} . But note also the Z/48. Our main question is the extent to which it gives rise to a fiber homotopy invariant of topological R"-bundles. The general result is
T h e o r e m I: Let ~, ¢ be two stabte R'~-bundles over X , and suppose they are fiber homotopy equivaient. Then there is a E H2(X; Z/2) and
24a 2 + PI(~) + 24{KS(~)} = P1(¢) + 24{KS(%b)}
in H4(X; Z/4S) where PI(() is the Z/48 reduction o[ the t~rst Pontraja~in c ~ s .
In other words, there is an element A • H4(BTSG; Z/48) with f*(A) = PI + 24{KS}, and (I) gives the effect of different liftings of a map po "g : X ----* BSTOP , BSG on A.
H2(BsToP; Z/2) = Z/2 with generator w2, so the possible factorizations of P0 through BTSG differ in their effect on A only by 24w~. In particular this gives
C o r o l l a r y : I f M 4 is a compact dosed topological manifold with even index, and u is its stable normal bundle, then wg = 0 • H2(M; Z/2) and
v ' f * (A) = PI(u) + 24{KS(~)}
is independent of the choice o f f factoring po.
This note came about in answer to a question of Frank Quinn. He pointed out that in [M-M l the exact structure of BSTOP, and the various surgery maps in dimension 4 were never worked out. But currently it appears very useful to understand them. Of course, we do not a t t empt to work out explicit geometric methods for evaluating the new invariants. But knowing what they are and how they fit together should make that fairly direct.
248
T h e h o m o t o p y t y p e s o f Bso, Bsa in d i m e n s i o n <_ 7
A Postnikov system for Bso through dimension 7 is given by
(1) Bso ,K(Z/2, "2.) ,K(Z, 5)
withK- invar ian t2{Sq2Sql ( t2)+t2 .Sql ( t2)} . (Note that HS(K(Z/2, 2); Z) = Z / 4 w i t h generator having mod(1) reduction "f and
(2) "7 = Sq2Sql( t2)+t2"Sq ' (12)-
Moreover, fl4(t~) = 7-) The stable homotopy of spheres is given in the first 6 dimensions by
(3) 7r,~(S °) =
Z i = 0 Z/2 i = 1, generator 77 Z/2 i = 2, generator '~1 Z/24 i = 3, generator v 0 i = 4 , 5 Z/2 i = 6, generator '~2 = v 2
and we will use the same names for the corresponding elements in ~ri+1(Bsa) ~ Try(S°). One relation that should be kept in mind is r}~l = 12v, since it also holds in rc.(BsG), though the relation q2 = ~1 which holds stably does not hold in ~r.(Bsa).
L e m m a (4): A Postnikov system for Bsc through 7 is given by
K ( Z / 2 , 2) × K(Z / 2 , 3) × K(Z/2, 7) ,K(Z/24 , 5)
where the K-invariant is 2{Sq2Sqa(t2) + t2 " Sql(t2)} + 4{,-,C'q2(ta)}.
P r o o f : With Z/24-coefficients the K-invariant for Bso maps back to the image of the cor- risponding K-invariant for Bso. Hence, the class in (2) must appear in the K-invariant. Also, the kernel of the map HS(K(Z/2, 2,3); Z/24) ~ HS(K(Z/2, 3); Z/24) is gen- erated by 4Sq2(ts). It follows that 4Sq2(ts) is the only term which can be added to the K-invariant. But, in fact, this term must be involved in the K-invariant because there is the homotopy relation which we have already noted qt;1 = 12v, since 7/is detected by Sq 2.
In order to understand the integral homology of Bso, BSTOP, and the intermediate space BTSG which we will introduce shortly, we need a method for obtaining Bochstein information from K-invariants. The following result will suffice.
,,,¢
L e m m a (5): Let K ( Z / 2 ' , j ) × K ( Z / 2 , j + 1 ) - - K ( Z / 2 " , j + 1) be given with
= 2",~(~j) + 2"-~(~i+~),
then tile fiber E of the map ~: is K ( Z / 2 i + ' - u ' - I × Z/2~').
P r o o f : The homotopy exact sequence of the fibration in dimensions j , j + 1 is
(6) 0 ,.,+~(E) ,Z/2 ,,Z/2' , . , (E) ,Z/2~. ,0
249
But the term 2"-ILj+~ in a*(~j+1) implies that g. is injective in (6). Thus E is a K 0 r , j ) and r is given as an extension in the sequence
o , z / 2 "-~ ,~j(E) ,Z/2'----,0.
The type of this extension is determined by the term 2~(~(, j)) in ~*(Lj+~). From this (5) follows.
(4) and (5) imply that there is a mod(8) Bochstein
/3s(~) = {Sq2(~a)} in H*(Bsc; Z/2).
Additionally, the Hurewicz image of z, is {w~} + 2{tg"} since this is already true in Bso, where it is well known. As a consequence H4(Bsa; Z) = Z/2 ~ Z/24 with generators {w~}, {w~'*} respectively, and 12u is in the kernel of the Hurewicz map.
T h e s t r u c t u r e of BSTOP t h r o u g h d i m e n s i o n 7
From the fiberings
(7) G/O , Bso , BsG
i l i G/TOP , BSTOP ' B s o
and the well known result of Kirby-Siebenmann that 7r4(G/TOP) = x4(G/O) = Z, but that the map between them is multiplication by 2, we get the diagram of extensions in a-4,
(8)
0 , Z
0 , Z
.24 Z
, ~ ( B s r o v )
Z/24 ,,, 0
:t , Z/24 ~ 0
The only way this diagram can commute is if ~r4(BsToP) = Z/2 @ Z with the element of order 2 mapping to 12 • v, and the generator of the Z-summand mapping to v.
Z/2 i = 2 L e m m a (9): zG(BSTOP) = Z (9 Z/2 i = 4
0 4 < i < 8 . BSTOP through this range is given by
Moreover, a Postnikov system/'or
(10) K(Z/2 , 2 ) x K(Z/2, 4) , K ( Z , 5)
with K-jnvariant 2{Sq2Sq~(~_) + ~2" SqI(L2)}.
(This is clear.)
In particular, the class {KS*} 6 H4(BsToP; Z) which is in the Hurewicz image of the element of order 2, must go to zero in H4(BsG; Z), since, in homotopy, it goes to 12v. This shows that {KS*} has no homology (or cohomology) relations implied by the
250
map into Bsa . However, in homotopy, the fact that it maps to 12u should have some consequeences.
The s p a c e BTSG
The failure to detect the l{{rby-Siebenmann class in H.(Bsa; Z) is the influence of the first exotic class L3. In fact, the term 4Sq2(ta) in the 5-dimensional K-invariant (4) is exactly the difficulty. (For example, if we kill u,2 but leave t3 in H*(Bs6; Z/2) the resulting space has only Z/4-torsion in H4( ; Z).) Hence it is natural to consider the classifying space BTSG obtained from B s c by killing the exotic class t3- For definiteness, recall that t3 is detected with 0-indeterminacy in the Thom-complex M S G by applying the twisted secondary operation associated to the relation (w2 + Sq2)(u,2 + Sq 2) to the Thorn class, and using the Thorn isomorphism to bring the class back to Bse . For details see [R].
We have the fibration sequence
(11). K ( Z / 2 , 2) ,BTsG ' BsG , K (Z /2 , 3)
with K-invariant L3. This is the universal space for fiber homotopy transversality to hold in the Thom space, at least through dimension 5 (Compare [B-M]). Indeed, a fiber homotopy sphere bundle ( ~ X and reduction to BTSG is equivalent to the condition ~3(() = 0 6 H3(X; Z/2), together with a specific choice of 2-dimensional cochain c so
where f : X - -~ B s a classifies (. This situation is very close, but certainly not the same as the situation studied in [F-K]. Also, there is a factorization of the canonical map BSTOP--*BsG as
BSTOP ~ BTSG , BSG.
Precisely, there are exactly two such factorizations differing by a map
BSTOP --~ K ( Z / 2 , 2).
Now, we look at the 6-skeleton of BTSG. This is the 6-skeleton of the 2-stage Postnikov system
K(Z/2 , 2 ) x K ( Z / 3 , 4) ,K(Z/8 , 5)
with K-invariant 2{Sq2Sq ~ (~2)+ ~" Sq~(~2)}. From (5) the resulting space has 4 ~h integral homology group given as
H4(BTsG; Z) : Z/2 (9 Z/48
with generators (w4)*, (w~)* respectively. Here, w~ can be identified with t2. Note that this implies that the Kirby-Siebenmann class maps non-triviMty to 24((w~)').
The proof of theorem (I)
f Lemma (12): Let X ~ B T s c be given and suppose f ' is the composite
(~,f) X , K(Z / 2 , 2) x BTS G ' BTSG
251
where # is the principal bundle map K(Z/2 , 2) × BTSG ~ BTSG, then
f"*{w~} = f'{w~.} +24a 2 6 H4(X; Z/48).
P roo f : H4(K(Z, 2) x BTSG; Z/16) = (Z/2) 2 $ Z/4 ~ Z/16 with generators
8(t2 ® w2), 8(1 ® w4) of order 2, (4t~ ® 1) of order 4, and (1 ®w~) of order 16.
We will show that > (w~) = 8(tr, ® 1) + 1 ® wa. We first note, by nat.urality and the primitivity of w~ in H4(Bso; Z) that 8(L.~ ® u,~.)" is not in this image. Next, we look at the cohomology Serre spectral sequence of the fibering
K(Z /2 , 2) :BTsG ~BsG
with Z/16-coefficients. E ° 4 = H4(K(Z/2,~; Z/16) = Z/4, with generator 4t~. Also, E~ o = H4(BsG; Z/16) = Z/2 • Z/8 with generators 8w4, 2(w2), and
E~ o = Hh(BsG; Z/16) = (Z/2) 3 + Z/8.
Here, only the Z/8 is of interest. It has generator Sq2(~), so d~(4~) = 4Sq2(t3), and at Eoo' ~, i + j = 4, only E ° 4 = Z/2, E 4 0 = Z/8 ~ Z/2 are non-zero. Thus there is a non-trivial extension for H4(BTSa; Z/16)
Theorem (I) is direct from (12). The corollary follows, also, since the assumption of even index implies that w2(M4) ~ = 0 (mod 2). Hence, either lifting gives the same map in cohomology with Z/48-coefficients.
C o n c l u d i n g r e m a r k s
From Quillen's work we know that BsG®Z2 can be identified with B(B+(SO(Fa))) in dimensions ~ 6, and as B(B+(Soo)) in all dimensions. Here, Sod is the infinite symmetric group. Similarly we can describe BTSG as B(B+(SO(F3))) in this same range. Moreover, BTSG can be given as B(B + (Sod)) in all dimensions. Here, these new groups are described by central extensions
Z/2 ,SO(F3) *SO(F3) ,0
z/2 , ~ ~s~ ,o
where, for Sod the extension is the (unique) non-trivial one for which the transposition (1, 2) continues to have order 2. This might be very useful in understanding Casson's recent results on the Rochtin invariant.
It seems direct to use the description above of BTSG by finite models to calculate the order of the classes which carry the remaining Pontrajagin classes. I hope to return to this later.
Also, there is a second factorizing space for the map BSTOP ~ Bsa, namely the space where we kill all the exotic classes cr(e 2,_1,2~_1). The precise structure of these classes is not entirely known, but there is considerable information in [R]. So it should
252
be possible to understand the higher torsion in the cohomology and homology of this intermediate classifying space. Moreover, it is likely that it is the universal space for the vanishing of transversality obstructions.
B i b l i o g r a p h y
[B-M] G. Brumfiel-J.Morgan, Homotopy theoretic consequences of N. Levitts obstruction theory to transver- sality for sphericad fibrations, Pac. J. Math (1976) 1-100
[F-K] M. Freedman-R. Kirby, A geometric proof of Rochlin's theorem, Algebraic and Geometric Topology, A.M.S. Proceedings of Symposia in Pure Mathematics, Vol. XXXII(1) (1978) 85-98
[M-M] Ib Madsen-R.J.Milgram, Classifying Spaces for Surgery and Cobordism of Manifolds, Ann. of Math Studies #92, Princeton U. Press (1979)
JR] Doug Ravenal, Thesis, Brandeis University (1970)
Suppose that X is a space with an action of the topological
group G. Let X G and X hG denote the fixed-point set respectively the homotopy fixed-point set of this action. We define
X hG := maPG(EG,X)
as the space of G-maps in the category Top of topological spaces and maps. As model for EG any acyclic G-complex is possible.
(Here complex always means CW-complex.) X hG is then unique up to homotopy.
The definition is not given in the category S of semisimplicial sets, as it happens in [DZ] and [M] for finite groups. For topological groups the space EG, constructed as nerve over a category, is not a simplicial set, but a semisimplicial object over the category Top . Therefore the same is true for the space
maPG(EG,X) ,
where X is interpreted as the singular chain complex of the topological space X. For finite groups both definitions agree up to weak homotopy [BK; chapter VIII ].
There are two other interpretations of the homotopy fixed-point set. The first one is as section space
U(EGXGG~BG)
of the fibration EGXGX ~ BG ,
the second one is as fixed-point set
map(EG,X) G ,
where G operates canonically on map(EG,X) . Let p be a prime, for all time fixed. X~ denotes the
E/p-completion in the sense of Bousfield and Kan [BK]. X is called E/p-good, if X~ is p-complete [BK; 1,5].
Especially nilpotent and other "nice" spaces are ~/p-good. Look at [BK;VII].
254
The unique G-map EG -~ * , where * is the one point set with a trivial G-action, induces a map
X G = maPG(*,X) , maPG(EG,X)
Functoriality of the composition gives a composite map
X G^ __~ X *G ~ X ~hG P P P
which fits into a commutative diagram
X G , X hG
1 l X G^ ~ X ~G
P P
Definition: A topological group N is called a p-toral group, iff there exists an exact sequence
1 --~ T --~ N --~ P --~ 1 ,
where T is a Torus and P a finite p-group.
Theorem: If N is a p-toral group and X a E/p-good connected finite N-complex, then the map
X N^ ~ X AhN P P
is a weak homotopy equivalence.
Remark: The analogue theorem for finite p-groups, but without the technical condition E/p-good, is proved by H. Miller in [M]. It is the foundation of the rest of the paper. For this result J. Lannes found another proof.
It is a pleasure to thank J. McClure for valuable discussions about the book of Bousfield and Kan.
2. Proof of the Theorem
We need some remarks: 2.1 Remark: Let
1 ~K ~G ,H ,I
be an exact sequence of topological spaces and assume, that H is finite. Let X be a G-space. H acts on the
255
fixed-point set X K canonically. We have
(X K) H = X G
As H is finite, EG is an acyclic K-complex of finite type. We get
X hK = map(EG,X) K ,
where map(EG,X) is a G-space. Hence using the above equation and the exponential law for mapping spaces, we get the analogue:
X hG ~ (xhK) hH
2.2 Remark: Let f:X 1 * X 2 be a weak homotopy equivalence
and a G-map between two G-spaces X I, X 2 The horizontal
map in the diagram
EGXGX 1 , EGXGX 2
\ Y BG
is a weak homotopy equivalence. Because BG is a complex, the two spaces
map(BG,EGXGX i) , i = 1 , 2
are weak homotopy equivalent as well.
We denote with map(BG,BG)
id
the connected component of the identity and with
map(BG,EGxGXi) s
the space of all maps, which are homotopic to a section. If we look at the two fibrations
hG • ~ map~BG,EGXGXi) s ' " X 1 b map(BG,BG)id f
it is easy to see, that the two homotopy fixed-point sets
X. hG are weak homotopy equivalent. 1
256
Proof of the theorem: i) reduction to the case of a torus. Let
1 tT ~N ~P tl
be the exact sequence belonging to th~ p-toral group N.
X T is a finite P-complex. It is proved in [M] that
X N^p = (xT) PAp ' (xTp) hP
is a weak homotopy equivalence. Setting
X~ hT = maPT(EN,X ~) ,
remark 2.1 implies a weak homotopy equivalence
x^hNp "w (Xp hT)hN
The map
X T^ ~ X ̂ h T P P
is P-equivariant. Together with (2.21 we can reduce therefore the problem to the case of a torus.
ii) Let n be the dimension of T. We can think of ~/pk c S 1
k as the group of the roots of unity with order p and define
Ok := (~/pk)n , O. := ~ O k
The homomorphism o. ---~ T induces a mod p-equivalence
Bc~ --~ T ,
which is the same as to say that the map
Hj(Bc.;~/p) , Hj(BT;~/p)
is an isomorphism.
Now let X be a ~/p-good connected finite T-complex. Then there are the following maps
hT ho. x; --, x;
T __ xO~ X
--~ lira X ~hOk , p
--~ tim X ck
As T is a finite Uk-complex for all k, X has also the
structure of a finite Ok-complex. Using Miller's Theorem
[M] and the following three propositions, the proof will be finished in a straightforward way.
257
2.3 Proposition: Let X be a finite T complex. Then it is
X T = i~XUk
and the sequence of the fixed-point sets is a finite sequence.
2.4 Proposition: Let X be a E/p-good finite T-complex. Then the map
X~ hT , X; ha-
is a weak homotopy equivalence.
2.5 Proposition: Let X be a finite T-complex. Then it is
h a . ha . ~n(X ~ k) "n (X~ ) ~i,
3. Proofs of the Propositions 2.3 - 2.5
Proof of 2.3: If X is a finite T-complex, it consists of a finite number of cells of the form T/A x e , where AcT is a closed subgroup, n
T/Axe n belongs to X T if A=T and it belongs to X uk
if UkCA . Because a~ is dense in T and because there is
only a finite number of orbit types T/A, we get
X ok = X T
for k big enough.
3.1 Lemma: Let Y be a p-complete space. Let
Y • E ~ B
be a fibration, such that the action of ~i B on
Hn(Y;E/p) is nilpotent. Then the E/p-completion
induces a homotopy equivalence
between the section spaces. Proof: Under the above assumption the mod R fibre lemma [BK; II, 5] is applicable. We get a Eibre square
258
E I E *
B ~ B A P
w h i c h i n d u c e s a c o m m u t a t i v e d i a g r a m
map(B,E) s P map(B,B)id
map B~,E~) s , map(B~,B~)id
where the rows are fibrations and the columns are given by the completion. With the universal property of pullback diagrams, which fibre squares are, you can prove, that (**) is up to homotopy a fibre square too. The fibres of the rows in (**) are exactly the section spaces. This implies the Lemma.
3.2 Lemma: Let E. p B. , i=0,1 , be fibrations with 1 1
p-complete fibre, in such a way that the diagram
i 0 --~ il
B 0 --* B 1
is a fibre square. Assume that the operation of ~I(BI)
on Hj(EI;~/p) is nilpotent and that the map B 0 ~ B 1
is a mod p equivalence. Then the two section spaces
F(E 1 -~ B I) , F(E 0 -~ B 0)
are weak homotopy equivalent. Proof: The assumptions of the mod R fibre square lemma [BK; II, 5.3 ] are satified. We get up to homotopy a fibre square
E0p P Elp
1 1 % ; '
with homotopy equivalences in the rows. This implies that the associated section spaces of the fibre squares are weak homotopy equivalent. If you use 3.1, the proof will be finished.
Proof of 2.4: The diagram
259
Ec~xa X p --~ ETXTX p
BC~ --~ BT
is a fibre square with a p-complete fibre in the columns. Moreover BT is 1-connected. Lemma 3.2 applies.
3.3 Lemma: Let GlCG2c... be a ascending sequence of groups
and define G. := ~ G k . Let X be a G.-space. Then the map
X hG- ~ hl~!~m xhGk
is a weak homotopy equivalence.
Proof: For the definition of holim see [BK; XI].
We choose the Milnor model for the spaces EGw and EG k. Then EG. is exactly the union of the spaces EG k or
EG~ = ~ EG k .
This implies that
hG~ = xhGk X = maPG (EG ,X) = ~ maPGk(EGk,X)
xhGk On the other hand the maps ~ X hGk-I are fibrations. According to [BK; XI] there is a weak homotopy equivalence
EG. = lim X hGk hG k , holim X e----
Proof of 2.5: Because of Lemma 3.3 there is an exact sequence
(Note that H*(MG;~(p)) can not have p-torsion since H°dd(MG;2Z/p) : 0
s. e.g. [3], VII (2.2))
Tensoring this morphism with ZZ/p gives an embedding
H~(M;~Z/p) @ ZZ/p[t] --~ H*(MG;ZZ/p) ® Zg/p[t] such that the cokernel is
ZZ/p [ t l-torsion.
Since A*p ~ H*(M;ZZ/p) is g-rigid it now follows, that M G ~ M induces
268
isomorphism H~(M;~) ~ H~(MG;~) and hence an we get:
Theorem 2: There exist simply-connected, orientable, closed 6-dimension-
al differentiable manifolds M such that for any closed orientable mani-
fold ~ with H*(M;~) ~ M*(M;~) a non-trivial action of ~ on M is only
possible for at most a finite number of primes p.
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