Equivariant Cohomological Chern Characters

More documents

Recommendations

Info

Sometimes also the following axiom is required.• Disjoint union axiomLet∐{X i | i ∈ I} be a family of G-CW -complexes. Denote by j i : X i →i∈I X i the canonical inclusion. Then the map( )∏∐HG(j n i ): HGn ∼ =X i −→ ∏ HG(X n i )i∈Ii∈I i∈Iis bijective.If HG ∗ is defined or considered only for proper G-CW -pairs (X, A), we callit a proper G-cohomology theory HG ∗ with values in R-modules.The role of the disjoint union axiom is explained by the following result. Itsproof for non-equivariant cohomology theories (see for instance [16, 7.66 and7.67]) carries over directly to G-cohomology theories.Lemma 1.1. Let H ∗ G and K∗ Gbe (proper) G-cohomology theories. Then(a) Suppose that HG ∗ satisfies the disjoint union axiom. Then there existsfor every (proper) G-CW -pair (X, A) with an exhaustion by subcomplexesA = X −1 ⊆ X 0 ⊆ X 1 ⊆ . . . ⊆ ⋃ n≥−1 X n = X a natural short exactsequence0 → lim 1 n→∞H p−1G(X n∪A, A) → H p (X, A) → limn→∞ Hp G (X n∪A, A) → 0;(b) Let T ∗ : HG ∗ → K∗ G be a transformation of (proper) G-cohomology theories,i.e. a collection of natural transformations T n : HG n → Kn G of contravariantfunctors from the category of (proper) G-CW -pairs to the categoryof R-modules indexed by n ∈ Z which is compatible with the boundaryoperator associated to (proper) G-CW -pairs. Suppose that T n (G/H) isbijective for every (proper) homogeneous space G/H and n ∈ Z.Then T n (X, A): HG ∗ (X, A) → K∗ G (X, A) is bijective for all n ∈ Z providedthat (X, A) is relative finite or that both H ∗ and K ∗ satisfy the disjointunion axiom.Remark 1.2 (The disjoint union axiom is not compatible with −⊗ Z Q).Let H ∗ G be a G-cohomology theory with values in Z-modules. Then H∗ G ⊗ Z Q isa G-cohomology theory with values in Q-modules since Q is flat as Z-module.However, even if H ∗ satisfies the disjoint union axiom, H ∗ G ⊗ Z Q does not satisfythe disjoint union axiom since − ⊗ Z Q is not compatible with products overarbitrary index sets.4
Example 1.3 (Rationalizing topological K-theory). Consider for instancethe (non-equivariant) cohomology theory with values in Z-modules satisfyingthe disjoint union axiom given by topological K-theory K ∗ . Let K ∗ (−; Q)be the cohomology theory associated to the rationalization of the K-theoryspectrum. This is a (non-equivariant) cohomology theory with values in Q-modules satisfying the disjoint union axiom. There is a natural transformationT ∗ (X): K ∗ (X) ⊗ Z Q → K ∗ (X; Q)of (non-equivariant) cohomology theory with values in Q-modules. The Q-mapT n ({pt.}) is bijective for all n ∈ Z. Hence T n (X) is bijective for all finiteCW -complexes by Lemma 1.1 (b). Notice that K ∗ (X) ⊗ Z Q does not satisfythe disjoint union axiom in contrast to K ∗ (X; Q). Hence we cannot expectT n (X) to be bijective for all CW -complexes. Consider the case X = BG fora finite group G. Since H p (BG; Q) ∼ = H p ({pt.}; Q) for all p ∈ Z, one obtainsK p (BG; Q) ∼ = K p ({pt.}; Q) for all p ∈ Z. By the Atiyah-Segal CompletionTheorem K p (BG) ⊗ Z Q ∼ = K p ({pt.}) ⊗ Z Q is only true if and only if the finitegroup G is trivial.Let α: H → G be a group homomorphism. Given an H-space X, define theinduction of X with α to be the G-space ind α X which is the quotient of G × Xby the right H-action (g, x) · h := (gα(h), h −1 x) for h ∈ H and (g, x) ∈ G × X.If α: H → G is an inclusion, we also write ind G H instead of ind α .A (proper) equivariant cohomology theory H? ∗ with values in R-modules consistsof a collection of (proper) G-cohomology theory HG ∗ with values in R-modules for each group G together with the following so called induction structure:given a group homomorphism α: H → G and a (proper) H-CW -pair(X, A) such that ker(α) acts freely on X, there are for each n ∈ Z naturalisomorphismsind α : H n G(ind α (X, A))∼ =−→ H n H(X, A) (1.4)satisfying(a) Compatibility with the boundary homomorphismsδ n H ◦ ind α = ind α ◦δ n G ;(b) FunctorialityLet β : G → K be another group homomorphism such that ker(β ◦ α) actsfreely on X. Then we have for n ∈ Zind β◦α = ind α ◦ ind β ◦H n K(f 1 ): H n H(ind β◦α (X, A)) → H n K(X, A),where f 1 : ind β ind α (X, A) ∼= −→ ind β◦α (X, A), (k, g, x) ↦→ (kβ(g), x) is thenatural K-homeomorphism;(c) Compatibility with conjugation5
Page 1 and 2: Equivariant Cohomological Chern Cha
Page 3: The paper is organized as follows:1
Page 7 and 8: Example 1.7 (Equivariant K-theory).
Page 9 and 10: 2. Modules over a CategoryIn this s
Page 11 and 12: sends M to the kernel of the map∏
Page 13 and 14: injective. We show by induction ove
Page 15 and 16: Given a contravariant ROr(G, F)-mod
Page 17 and 18: We get a natural transformation ch
Page 19 and 20: y the composite∏p+q=n chp,q(X,A)H
Page 21 and 22: We want to use Theorem 2.14 (b) to
Page 23 and 24: h −1 im(g)h = im(f) and therefore
Page 25 and 26: if we defineT H (K q (BH)) :=⎛⎞
Page 27 and 28: the G-cohomology theory BHG ∗ . C
Page 29 and 30: References[1] P. Baum, A. Connes, a

Equivariant Cohomological Chern Characters

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?