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\(K\)-cycles for twisted \(K\)-homology. (English) Zbl 1300.19003
In the paper under review, the authors “propose a new approach to the construction of twisted geometric cycles for (locally finite) \(CW\)-complexes motivated by the study of \(D\)-branes in string theory”. A twisting datum on a second countable locally compact Hausdorff topological space \(X\) is a locally trivial bundle \(\mathcal A\) of elementary \(C^*\)-algebras on \(X\). A \(D\)-cycle for \((X,\mathcal A)\) is a quadruple \((M,E,\varphi,S)\) consisting of a closed oriented \(C^{\infty}\) Riemannian manifold \(M\), a complex vector bundle \(E\) on \(M\), a continuous map \(\varphi\) from \(M\) to \(X\), and a spinor bundle \(\mathcal S = \mathrm{Cliff}^+_\mathbb{C}(TM) \otimes \varphi^* \mathcal A^{\mathrm{op}}\). Under the equivalence classes of \(D\)-cycles there are natural operations like bordance, direct-sum union. The group of \(K^{\mathrm{geo}}_*(X,A)\) with respect to the disjoint union of \(D\)-cycles is mapped by the twisted index map \(\mu\) to the Kasparov KK-group, \[ \mu : K^{\mathrm{geo}}_*(X,\mathcal A) \to KK^*(\Gamma(X,\mathcal A), \mathbb C). \] There is a natural surjective map \(\Psi : K^{\hom}(X,\mathcal P_\alpha) \twoheadrightarrow K^{\mathrm{geo}}(X,\mathcal P_\alpha)\) and an isomorphism \(\Phi : K^{\hom}(X,\mathcal P_{\alpha}) \to KK^*(\Gamma(X,\mathcal P_\alpha(\mathcal K(H))), \mathbb C)\).
The group \(K^{\mathrm{top}}_*(X,\mathcal A)_\Lambda = K^{\mathrm{top}}_0(X,\mathcal A)_\Lambda \oplus K^{\mathrm{top}}_1(X,\mathcal A)_\Lambda\) of equivalence classes of properly supported \(K\)-cycles \((M,\varphi,\sigma)\) consisting of a \(\mathrm{Spin}^c\) manifold without boundary, a continuous proper map \(\varphi: M \to X\), and an element \(\sigma\) in the representative \(K_0\)-theory of \(\Gamma_0(M,\varphi^* \mathcal A^{\mathrm{op}})\).
The main result of the paper is Theorem 3 stating that for a locally finite \(CW\)-complex \(X\) with a given twisting datum \(\mathcal A\), the natural map \[ \eta_X : K^{\mathrm{top}}(X,\mathcal A) \to KK^j_c(\Gamma_0(X,\mathcal A),\mathbb C) \] is an isomorphism of abelian groups.
In Theorem 7 the authors prove that for any locally finite, countable and finite dimensional \(CW\)-complexes \(X\) with a given twisting datum \(\mathcal A\), the natural map \[ \eta_X : K^{\mathrm{top}}(X,\mathcal A)_\Lambda \to KK^j(\Gamma_0(X,\mathcal A),\mathbb C) \] is an isomorphism of abelian groups.

MSC:
19L50 Twisted \(K\)-theory; differential \(K\)-theory
19K35 Kasparov theory (\(KK\)-theory)
Biographic References:
Kuku, Aderemi
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