a product in equivariant homology for compact lie …htene/tene...a product in equivariant homology...

IntroductionThe product in Tate cohomology

A product in equivariant homologyproperties and computations

A product in equivariant homologyfor compact Lie group actions

Haggai Tene

University of Bonn

September 3, 2014

Haggai Tene A product in equivariant homology



Introduction

In this talk, I am going to discuss a product in (Borel) equivarianthomology for compact Lie group actions. This product is relatedto the cup product in negative Tate cohomology.

The talk is structured as follows:

1 Tate cohomology and how does it give a product inequivariant homology.

2 An explicit construction of a product by homotopy-theoreticmeans and its relation to the product in Tate cohomology.

3 Some results and some computations regarding this product.

This is a joint work with S. Kaji.




Tate cohomology

For a finite group G there is a certain duality which makes itpossible incorporate group homology and group cohomology intoone object - Tate cohomology. For a Z[G ] module P we will havethe following isomorphism (n 6= 0,−1):

Hn(G ;P) =

{Hn(BG ;P) : n ≥ 1H−(n+1)(BG ;P) : n ≤ −2

For a ring R, Tate cohomology is multiplicative, which induces aproduct in the homology of BG with a dimension shift:

Hk(BG ;R)⊗ Hl(BG ;R)→ Hk+l+1(BG ;R)

(where k , l > 0).




Tate cohomology

This product was studied by Benson and Carlson (with coefficientsin a field K of characteristic p). They showed that in many casesthis product vanishes.

Theorem [Benson-Carlson]

Suppose that the p-rank of G is two or more. If H∗(G ;K ) isCohen-Macaulay then the product vanishes.

This is the case, for example, if there is a p-Sylow subgroup withnon cyclic center.

There are two extremes. On the one hand periodic groups, wherethe product is highly non trivial. On the other hand the groupswhich appear in the theorem with vanishing products. In betweenit is not known, though there are non periodic groups with nontrivial products.




Tate cohomology

This product was studied by Benson and Carlson (with coefficientsin a field K of characteristic p). They showed that in many casesthis product vanishes.

Theorem [Benson-Carlson]

Suppose that the p-rank of G is two or more. If H∗(G ;K ) isCohen-Macaulay then the product vanishes.

This is the case, for example, if there is a p-Sylow subgroup withnon cyclic center.There are two extremes. On the one hand periodic groups, wherethe product is highly non trivial. On the other hand the groupswhich appear in the theorem with vanishing products. In betweenit is not known, though there are non periodic groups with nontrivial products.




Tate cohomology

Tate cohomology was generalized to an equivariant cohomologytheory first by Swan (for finite groups) and later by Greenlees andGreenlees-May (for compact Lie groups). This theory still enjoyscertain duality: If G is a compact Lie group acting smoothly on aclosed oriented manifold M, then under some appropriateconditions we have the following isomorphism:

HnG (M;R) =

{HnG (M;R) : n ≥ dim(M) + 1

HGdim(M)−dim(G)−n−1(M;R) : n ≤ −2

Remark: HGk (M;R) stands for the homology of the Borel construction.




Tate cohomology

HnG (M;R) =

{HnG (M;R) : n ≥ dim(M) + 1

HGdim(M)−dim(G)−n−1(M;R) : n ≤ −2

The cup product in this case induces a product in equivarianthomology with a certain shift in degree:

HGk (M;R)⊗ HG

l (M;R)→ HGk+l+dim(G)+1−dim(M)(M;R)

(where k , l > dim(M)− dim(G )).

What is this product?




Kreck’s product

Kreck defined a product on H∗(BG ;Z) when G is a compact Liegroup (with a certain restriction). This was done in a geometricway, using stratifolds and the join operation.

Theorem [T]

In case G is finite, Kreck’s product coincides with the product innegative Tate cohomology, under the identification

Hk(BG ;Z)∼=−→ H−k−1(G ;Z) for k > 0.

We give a generalization of this construction and define a productin equivariant homology. Instead of using Kreck’s geometricmethods we use a homotopy-theoretic construction.




A product in equivariant homology

We start by describing an external product in HG∗ . The product we

describe is a secondary product, and as such, it is only defined forG -spaces of finite homological dimension and for classes above acertain degree. Later, using the equivariant Gysin map, we definean internal product for the case of a smooth action on a manifold.





G - compact Lie group.X ,Y - G spaces of finite homological dimensions.

XG ,YG - the Borel construction.

P → T → T↓ ↓ ↓S → X ×G Y → YG

↓ ↓ ↓S → XG → BG

HGk (X ;R)⊗ HG

l (Y ;R) ∼= HGk (S ;R)⊗ HG

l (T ;R)×−→ Hk+l(S × T ;R)→ Hk+l(S × T ;Hdim(G)(G ))→ Hk+l+dim(G)(P;R)

(The last step uses the Serre SS for the fibration G ↪→ P → S × T )







P → T → T↓ ↓ ↓S → X ×G Y → YG

↓ ↓ ↓S → XG → BG

HGk (X ;R)⊗ HG

l (Y ;R) ∼= Hk(S ;R)⊗ Hl(T ;R)×−→ Hk+l(S × T ;R)

→ Hk+l(S × T ;Hdim(G)(G ))→ Hk+l+dim(G)(P;R)

(The last step uses the Serre SS for the fibration G ↪→ P → S × T )







P → T → T↓ ↓ ↓S → X ×G Y → YG

↓ ↓ ↓S → XG → BG

HGk (X ;R)⊗ HG

l (Y ;R)→ Hk+l+dim(G)(P;R)

Composing with the following map is trivial:

Hk+l+dim(G)(P;R)→ Hk+l+dim(G)(X ×G Y ;R)







P → T → T↓ ↓ ↓S → X ×G Y → YG

↓ ↓ ↓S → XG → BG

HGk (X ;R)⊗ HG

l (Y ;R)→ Hk+l+dim(G)(P;R)Composing with the following map is trivial:

Hk+l+dim(G)(P;R)→ Hk+l+dim(G)(X ×G Y ;R)





We construct the homotopy pushout:

P //

��

T

��

zzB

q

$$S //

@@

X ×G Y .

We examine the Mayer Vietoris sequence associated to it. DenoteN = k + l + dim(G ):

· · · → HN+1(S;R)⊕ HN+1(T ;R)→ HN+1(B;R)∂−→ HN(P;R)→ HN(S ;R)⊕ HN(T ;R)→ · · · .





We define the product to be the composition:

HGk (X ;R)⊗ HG

l (Y ;R)→ Hk+l+dim(G )(P ;R)

∂−1

−−→ Hk+l+dim(G )+1(B ;R)→ HGk+l+dim(G )+1(X×Y ;R)




An internal product

If X = Y and there is an equivariant map X × X → X then oneobtains a pontryagin product. An interesting and non trivialexample is when X = G ad , the group G with the conjugationaction. In this case we obtain a product in the homology of thefree loop space on G , given by the homotopy equivalence

LBG ∼ EG ×G G ad .




An internal product

Suppose now that M is a smooth manifold with a smooth andorientation preserving action of G . We can define an internalproduct in HG

∗ (M;R) by composing the external product with theequivariant Gysin map associated to the diagonal inclusion

∆ : M ↪→ M ×M

The grading is given by:

HGk (M;R)⊗ HG

l (M;R)→ HGk+l+dim(G)−dim(M)+1(M;R)

when k, l > dim(M)− dim(G ).




An internal product

One might check that the product is obtained from the followingpullback-pushout diagram:

P //

��

T

��

}}B

q

!!S //

??

MG .




Examples

The simplest example is when M = pt and k = l = 0 (here wehave to assume that dim(G ) > 0).

One can compute the productusing the following pullback-pushout diagram:

G //

��

pt

��

||ΣG

q

""pt //

==

BG ,

One can follow the construction and see that the product of thegenerator with itself is given by the image of a fundamental classof ΣG . For G = S1 this gives a generator of H2(BS1).




Examples

The simplest example is when M = pt and k = l = 0 (here wehave to assume that dim(G ) > 0). One can compute the productusing the following pullback-pushout diagram:

G //

��

pt

��

||ΣG

q

""pt //

==

BG ,

One can follow the construction and see that the product of thegenerator with itself is given by the image of a fundamental classof ΣG . For G = S1 this gives a generator of H2(BS1).




Examples

Another example is when M = pt and G = S1 (or similarly Gfinite cyclic or SU(2)), then the product is given by the followingdiagram:

(S2k+1 × S2l+1)/S1 //

��

CP l

��

yyCPk+l+1

q

%%CPk //

66

CP∞,

One sees that the product of generators in degree 2k and 2l is agenerator in degree 2k + 2l + 2.




Naturality

The product is natural with respect to the following maps:

Restriction and inflation for H ≤ G .

Gysin maps for smooth maps between manifolds.




Computations for Lie groups

Theorem

Assume G = G1 × G2 acting diagonally on M = M1 ×M2. Setk = k1 + k2 and l = l1 + l2, if k1, l1 > dim(M1)− dim(G1) andl2 > dim(M)− dim(G ), then the following composition vanishes:

(HG1k1

(M1;R)⊗ HG2k2

(M2;R))⊗(HG1l1

(M1;R)⊗ HG2l2

(M2;R))

↓ ×HGk (M;R)⊗ HG

l (M;R)↓ ∗

HGk+l+dim(G)−dim(M)+1(M;R).





Restricting to the case M = pt, this implies the vanishing of theproduct in H∗(BG ,Z) for many groups G :

Corollary

The product vanishes for the torus T n for n > 1.

Proposition

The product vanishes for U(n) and Sp(n) when n > 1, for SU(n)when n > 2 and for SO(n) when n > 3.

Proposition

Rationally, the product vanishes for all groups of rank > 1.





This is not the case when the rank is 1. Here G is one of thefollowing:

U(1) - H2k(BU(1),Z) ∼= Z〈a2k〉 and a2k ∗ a2l = a2k+2l+2.SU(2) - H4k(BSU(2),Z) ∼= Z〈a4k〉 and a4k ∗ a4l = a4k+4l+4.SO(3) - H∗(BSO(3);Z) ∼= ⊕Z〈b4k〉 ⊕Z/2Z〈yi ,j〉, (i , k ≥ 0, j > 0)

b4k ∗ b4l = 2b4k+4l+4 and all other products are trivial.





For the case G = SO(3) we use the following:

Proposition

Let φ : G → G ′ be a surjective homomorphism with finite kernel oforder n, then for the induced map: φ∗ : H∗(BG ,Z)→ H∗(BG

′,Z)we have:

φ∗(α ∗ β) = n · φ∗(α) ∗ φ∗(β)


a product in equivariant homology for compact lie …htene/tene...a product in equivariant homology...

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