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Topological $$K$$-(co)homology of classifying spaces of discrete groups. (English) Zbl 1262.55002
This paper presents a method for computing generalized cohomology and homology of the Borel construction $$EG\times_GX$$ on $$X$$, where $$G$$ is a discrete group and $$X$$ is a proper $$G$$-CW-complex. If we focus on the case of a cohomology theory $$\mathcal{H}^*$$, then the main result of this paper is Theorem 3.6, in particular its last assertion (iii), which states that under some conditions for appropriate natural numbers $$r^k_p(X)$$ there is an exact sequence $0 \to A \to \mathcal{H}^k(G\setminus X) \to \mathcal{H}^k(EG\times_GX) \to B\times \prod_{p \in \mathcal{P}(X)}(\mathbb{Z}^{hat{ \;}}_p)^{r^k_p(X)} \to C \to 0$ where $$A$$, $$B$$ and $$C$$ are finite abelian groups with $A\otimes_{\mathbb{Z}}\mathbb{Z}\left[\frac{1}{\mathcal{P}(X)}\right] =B\otimes_{\mathbb{Z}}\mathbb{Z}\left[\frac{1}{\mathcal{P}(X)}\right] =C\otimes_{\mathbb{Z}}\mathbb{Z}\left[\frac{1}{\mathcal{P}(X)}\right]=0.$ Here $$\mathcal{P}(X)$$ denotes the set of primes $$p$$ which divides $$|G_x|$$ for $$x \in X$$. As a particular application of the approach used to get the exact sequence above and its dual, the topological $$K$$-theory of $$BG$$ is discussed. Let $$X$$ be a finite proper $$G$$-CW-complex satisfying the condition $$\tilde{H}_k(X)=0$$, $$k \in \mathbb{Z}$$, then it is shown that the above exact sequence can be transformed into an exact sequence $0 \to A \to K^k(G\setminus X) \to K^k(BG) \to B\times \prod_{p \in \mathcal{P}(G)}(\mathbb{Z}^{\hat{ \;}}_p)^{r^k_p(G)} \to C \to 0$ where $$\mathcal{P}(G)$$ is given as the set of primes $$p$$ dividing $$|H|$$ of some finite subgroup $$H \subset G$$ and $$r^k_p(G)$$ is given by $r^k_p(G)=\sum_{(g) \in \text{con}_p(G)}\sum_{i \in \mathbb{Z}}\text{dim}_{\mathbb{Q}} H^{k+2i}(BC_G(g); \mathbb{Q}),$ $$\text{con}_p(G)$$ denoting the set of conjugacy classes of elements of $$p$$-power order in $$G$$. Moreover the authors prove universal coefficient theorems for equivariant $$K$$-theory and a cocompletion theorem for equivariant $$K$$-homology in connection with the proof of this $$K$$-theory exact sequence. Finally they give examples of computations which follow directly from assertion (i) of Theorem 3.6, though assertion (iii) also follows from this.
##### MSC:
 55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology 55N15 Topological $$K$$-theory 19L47 Equivariant $$K$$-theory
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