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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.

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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