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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)
##### Keywords:
twisted $$K$$-theory; $$D$$-branes