Equivariant Cohomological Chern Characters
Equivariant Cohomological Chern Characters
Equivariant Cohomological Chern Characters
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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