Equivariant Cohomological Chern Characters

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This product is required to be compatible with the boundary homomorphism ofthe long exact sequence of a pair, to be graded commutative, to be associativeand to have a unit 1 ∈ H 0 ({pt.}).Given a multiplicative structure on H ∗ , we obtain for every p, q ∈ Z a pairing∪: H q ({pt.}) ⊗ R H q′ ({pt.}) → H q+q′ ({pt.}).It yields on singular (or equivalently cellular) cohomology a productH p (X, A; H q ({pt.}))⊗ R H p′ (X, B; H q′ ({pt.})) → H p+p′ (X, A∪B; H q+q′ ({pt.})).The collection of these pairings induce a multiplicative structure on the cohomologytheory given by ∏ p+q=n Hp (X, A; H q ({pt.})). The straightforwardproof of the next lemma is left to the reader.Lemma 6.2. Let R be a commutative ring with Q ⊆ R. Let H ∗ be a (nonequivariant)cohomology theory satisfying the disjoint union axiom which comeswith a multiplicative structure.Then the (non-equivariant) <strong>Chern</strong> character of Example 4.1ch n (X, A): H n (X, A) ∼= −→∏p+q=nH p (X, A, H q (∗))is compatible with the given multiplicative structure on H ∗ and the induced multiplicativestructure on the target.Next we deal with the equivariant version. We only deal with the propercase, the definitions below make also sense without this condition.Let HG ∗ be a proper G-cohomology theory. A multiplicative structure assignsto a proper G-CW -complex X with G-CW -subcomplexes A, B ⊆ X natural R-homomorphisms∪: H n G(X, A) ⊗ R H n′G (X, B) → H n+n′G(X, A ∪ B). (6.3)This product is required to be compatible with the boundary homomorphismof the long exact sequence of a G-CW -pair, to be graded commutative, to beassociative and to have a unit 1 ∈ HG 0 (X) for every proper G-CW -complex XLet H? ∗ be a proper equivariant cohomology theory. A multiplicative structureon it assigns a multiplicative structure to the associated proper G-cohomologytheory HG ∗ for every group G such that for each group homomorphismα: H → G the maps given by the induction structure of (1.4)ind α : H n G(ind α (X, A))∼ =−→ H n H(X, A)are in the obvious way compatible with the multiplicative structures on HG ∗ andHH ∗ Ṅext we explain how a given multiplicative structure on H ?∗ induces oneon BH? ∗ . We have to specify for every group G a multiplicative structure on26
the G-cohomology theory BHG ∗ . Consider a G-CW -complex X with G-CW -subcomplexes A, B ⊆ X. For two contravariant ROr(G; F)-chain complexes C ∗and D ∗ define the contravariant ROr(G; F)-chain complexes C ∗ ⊗ R D ∗ by sendingG/H to the tensor product of R-chain complexes C ∗ (G/H) ⊗ R D ∗ (G/H).Leta ∗ : C∗ROr(G;F) (X, A) ⊗ R C∗ ROr(G;F) (X, B)∼ =−→ C∗ROr(G;F) ((X, A) × (X, B))be the isomorphism of ROr(G; F)-chain complexes which is given for an objectG/H by the natural isomorphism of cellular R-chain complexesC ∗ (X H , A H ) ⊗ R C ∗ (X H , B H ) ∼= −→ C ∗ ((X H , A H ) × (X H , B H )).The multiplicative structure on HG ∗ yields a map of contravariant ROr(G; F)-modulesc: H q G (G/?) ⊗ R H q G(G/?) → Hq+qG(G/?).Let∆: (X; A ∪ B) → (X, A) × (X, B), x ↦→ (x, x)be the diagonal embedding. Define a R-cochain map by the composite()b ∗ : hom ROr(G;F) C∗ ROr(G;F) (X, A), H q G (G/?)hom ROr(G;F)(⊗ R hom ROr(G;F)(C ROr(G;F)∗ (X, B), H q′G (G/?) )C∗ROr(G;F)hom ROr(G;F)((a ∗) −1 ,c)(−−−−−−−−−−−−−−−→ hom ROr(G;F)⊗R−−→(X, A) ⊗ R C ROr(G;F)∗ (X, B), H q G (G/?) ⊗ R H q′G (G/?) )C∗ROr(G;F)(hom ROr(G;F) (C ROr(G;F)∗ (∆),id)−−−−−−−−−−−−−−−−−−−−−→ hom ROr(G;F) C∗ROr(G;F)There is a canonical R-mapH ∗ (C ∗ ⊗ R D ∗ ) → H ∗ (C ∗ ⊗ R D ∗ )((X, A) × (X, B)), H q+q′G (G/?) )(X, A ∪ B), H q+q′G (G/?) ).for two R-cochain complexes C ∗ and D ∗ . This map together with the mapinduced by b ∗ on cohomology yields an R-homomorphismH p ROr(G;F) (X, A; Hq G (G/?)) ⊗ R H p′ROr(G;F)(X, B; Hq′G (G/?))→ H p+p′ROr(G;F) (X, A ∪ B; Hq G (G/?)) .The collection of these R-homomorphisms yields the desired multiplicative structureon BHG ∗ . We leave it to the reader to check that the axioms of a multiplicativestructure on BHG ∗ are satisfied and that all these are compatible with theinduction structure so that we obtain a multiplicative structure on the equivariantcohomology theory BH? ∗ .We also omit the lengthy but straightforward proof of the following resultwhich is based on Theorem 4.6, Lemma 6.2 and the compatibility of the multiplicativestructure with the induction structure.27
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 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: if we defineT H (K q (BH)) :=⎛⎞
Page 29 and 30: References[1] P. Baum, A. Connes, a

Equivariant Cohomological Chern Characters

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