zbMATH — the first resource for mathematics

\(K\)-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.

19L50 Twisted \(K\)-theory; differential \(K\)-theory
19K35 Kasparov theory (\(KK\)-theory)
Biographic References:
Kuku, Aderemi
Full Text: DOI
[1] Algebraic Topology (1980) · Zbl 0435.55001
[2] New York J. Math. 11 pp 387– (2005)
[3] Springer, Lecture Notes in Mathematics 738 · Zbl 1177.00007
[4] Journal of K-Theory 1 pp 357– (2008)
[5] Differential Forms in Algebraic Topology (1982) · Zbl 0496.55001
[6] Quanta of Maths 11 (2010)
[7] Pure and Applied Mathematics Quarterly 3 pp 1– (2007) · Zbl 1146.19004
[8] Proc. Symp., Pure Math. 38 pp 117– (1982)
[9] Proc. Symp. Pure Math. 81 pp 81– (2010)
[10] Spin Geometry (1989) · Zbl 0688.57001
[11] DOI: 10.1016/0040-9383(65)90067-4 · Zbl 0129.38901
[12] The Novikov Conjecture – Geometry and Algebra (2005) · Zbl 1058.19001
[13] DOI: 10.1007/BF01404917 · Zbl 0647.46053
[14] Izv. Acad. Nauk SSSR Ser. Mat. 44 pp 571– (1980)
[15] Acad. Nauk SSSR Ser. Mat. 39 pp 796– (1975)
[16] DOI: 10.1090/conm/265/04244
[17] DOI: 10.1007/s00208-007-0171-6 · Zbl 1147.19006
[18] DOI: 10.1017/is011010022jkt170 · Zbl 1256.19005
[19] Elements of Homotopy Theory (1978) · Zbl 0406.55001
[20] Index theory and twisted K-homology J. Noncommut. Geom. 2 pp 497– (2008)
[21] Trans. Amer. Math. Soc. 361 pp 1269– (2009)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.