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Equivariant-bivariant Chern character for profinite groups. (English) Zbl 0997.55010
Let $$G$$ be a totally disconnected group, and let $$X$$ and $$Y$$ be locally compact spaces on which $$G$$ acts. The authors define a bivariant, $$\mathbb{Z}$$-graded cohomology theory $$H_{G,c}^{*}(Y,X)$$ using sheaf theoretic techniques. They show that these groups generalize some known equivariant cohomology groups previously known in special cases. Furthermore, they conjecture the existence of a Chern character morphism: $$ch: KK_{G}^{i} (C{0}(Y),C{0}(X))\to \bigoplus_{n\in\mathbb{Z}} H^{j+2n}_{G,c}(Y,X)$$, $$j=0,1,$$ that becomes an isomorphism of $$\mathbb{C}$$ vector spaces when the domain of $$ch$$ is tensored (over $$\mathbb{Z}$$) with $$\mathbb{C}$$, where $$C_{0}(Z)$$ is the Abelian $$C^{*}$$ algebra of continuous complex valued functions on Z vanishing at infinity and $$KK^{*}_{G}$$ stands for the Kasparov equivariant bivariant $$K$$-theory. The authors prove their conjecture above in the case of having $$G$$ a profinite group. Finally, they also prove that $$H^{*}(pt,X)$$ is representable as the homotopy of a suitable function space when both $$G$$ and $$X$$ are compact.

##### MSC:
 55N25 Homology with local coefficients, equivariant cohomology 19K35 Kasparov theory ($$KK$$-theory) 19L47 Equivariant $$K$$-theory 19L10 Riemann-Roch theorems, Chern characters
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