Equivariant Cohomological Chern Characters

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is much simpler. The equivariant <strong>Chern</strong> character will identify this simplerproper equivariant cohomology theory with the given one.The orbit category Or(G) has as objects homogeneous spaces G/H and asmorphisms G-maps. Let Sub(G) be the category whose objects are subgroupsH of G. For two subgroups H and K of G denote by conhom G (H, K) theset of group homomorphisms f : H → K, for which there exists an elementg ∈ G with gHg −1 ⊆ K such that f is given by conjugation with g, i.e. f =c(g): H → K, h ↦→ ghg −1 . Notice that f is injective and c(g) = c(g ′ ) holdsfor two elements g, g ′ ∈ G with gHg −1 ⊆ K and g ′ H(g ′ ) −1 ⊆ K if and only ifg −1 g ′ lies in the centralizer C G H = {g ∈ G | gh = hg for all h ∈ H} of H in G.The group of inner automorphisms of K acts on conhom G (H, K) from the leftby composition. Define the set of morphismsmor Sub(G) (H, K) :=Inn(K)\ conhom G (H, K).There is a natural projection pr: Or(G) → Sub(G) which sends a homogeneousspace G/H to H. Given a G-map f : G/H → G/K, we can choosean element g ∈ G with gHg −1 ⊆ K and f(g ′ H) = g ′ g −1 K. Then pr(f)is represented by c(g): H → K. Notice that mor Sub(G) (H, K) can be identifiedwith the quotient mor Or(G) (G/H, G/K)/C G H, where g ∈ C G H actson mor Or(G) (G/H, G/K) by composition with R g −1 : G/H → G/H, g ′ H ↦→g ′ g −1 H.Denote by Or(G, F) ⊆ Or(G) and Sub(G, F) ⊆ Sub(G) the full subcategories,whose objects G/H and H are given by finite subgroups H ⊆ G. BothOr(G, F) and Sub(G, F) are EI-categories of finite length.Given a proper G-cohomology theory HG ∗ with values in R-modules we obtainfor n ∈ Z a contravariant ROr(G, F)-moduleHG n (G/?): Or(G, F) → R - MOD, G/H ↦→ Hn G (G/H). (3.1)Let (X, A) be a pair of proper G-CW -complexes. Then there is a canonicalidentification X H = map(G/H, X) G . Thus we obtain contravariant functorsOr(G, F) → CW -PAIRS, G/H ↦→ (X H , A H );Sub(G, F) → CW -PAIRS, G/H ↦→ C G H\(X H , A H ),where CW -PAIRS is the category of pairs of CW -complexes. If we composethem with the covariant functor CW -PAIRS → Z-CHCOM sending (Z, B)to its cellular Z-chain complex, then we obtain the contravariant ZOr(G, F)-chain complex C∗Or(G,F) (X, A) and the contravariant ZSub(G, F)-chain complexC∗Sub(G,F) (X, A). Both chain complexes are free in the sense that eachchain module is a free ZOr(G, F)-module resp. ZSub(G, F)-module. Namely,if X n is obtained from X n−1 ∪ A n by attaching the equivariant cells G/H i × D nfor i ∈ I n , thenC Or(G,F)n (X, A) ∼ =⊕Z mor Or(G,F) (G/?, G/H i ); (3.2)i∈I n⊕Cn Sub(G,F) (X, A) ∼ = Z mor Sub(G,F) (?, H i ). (3.3)i∈I n14
Given a contravariant ROr(G, F)-module M, the equivariant Bredon cohomology(see [2]) of a pair of proper G-CW -complexes (X, A) with coefficients in Mis defined by( )HOr(G,F) n (X, A; M) := Hn hom ZOr(G,F) (C∗ Or(G,F) (X, A), M) . (3.4)This is indeed a proper G-cohomology theory satisfying the disjoint union axiom.Hence we can assign to a proper G-homology theory HG ∗ another proper G-cohomology theory which we call the associated Bredon cohomology∏BHG(X, n A) := H p Or(G,F) (X, A; Hq G(G/?)). (3.5)p+q=nThere is an obvious ZSub(G; F)-chain mappr ∗ C∗Or(G,F) (X, A) −→ ∼= C∗ Sub(G,F) (X, A)which is bijective because of (3.2), (3.3) and the canonical identificationpr ∗ Z mor Or(G,F) (G/?, G/H i ) = Z mor Sub(G,F) (?, H i ).Given a covariant ZSub(G, F)-module M, we get from the adjunction (pr ∗ , pr ∗ )(see Lemma 2.13 (a)) natural isomorphismsHROr(G,F) n (X, A; res pr M)∼ ( ))=−→ H n hom ZSub(G,F)(C ∗Sub(G,F) (X, A), M . (3.6)This will allow us to work with modules over the category Sub(G; F) which issmaller than the orbit category and has nicer properties from the homologicalalgebra point of view. The main advantage of Sub(G; F) is that the automorphismgroups of every object is finite.Suppose, we are given a proper equivariant cohomology theory H?∗ withvalues in R-modules. We get from (3.1) for each group G and n ∈ Z a covariantRSub(G, F)-moduleHG n (G/?): Sub(G, F) → R - MOD, H ↦→ Hn G (G/H). (3.7)We have to show that for g ∈ C G H the G-map R g −1 : G/H → G/H, g ′ H →g ′ g −1 H induces the identity on HG n (G/H). This follows from Lemma 1.5.We will denote the covariant ROr(G, F)-module obtained by restriction withpr: Or(G, F) → Sub(G, F) from the RSub(G, F)-module HG n (G/?) of (3.7)again by HG n (G/?) as introduced already in (3.1).It remains to show that the collection of G-cohomology theories BHG ∗ (X, A)defined in (3.4) inherits the structure of a proper equivariant cohomology theory,i.e. we have to specify the induction structure. We leave it to the reader to carryout the obvious dualization of the construction for homology in [8, Section 3]and to check the disjoint union axiom.15
Page 1 and 2: Equivariant Cohomological Chern Cha
Page 3 and 4: The paper is organized as follows:1
Page 5 and 6: Example 1.3 (Rationalizing topologi
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: injective. We show by induction ove
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

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