K-cycles for twisted K-homology

Abstract

We summarise the construction of geometric cycles and their use in describing the Kasparov K-homology of a CW-complex X . When Kasparov Khomology is twisted by a degree three element of theČech cohomology of X then there is a corresponding construction of twisted geometric cycles for the case where X is a smooth manifold however the method that was employed does not apply in the case of CW-complexes. In this article we propose a new approach to the construction of twisted geometric cycles for CW-complexes motivated by the study of D-branes in string theory.

Relative geometric assembly and mapping cones, part I: the geometric model and applications

“…If A is unital, then an A-bundle refers to a locally trivial bundle with fibers given by finitely generated, projective Hilbert A-modules. An example of such a bundle is the Mischchenko bundle, L BΓ = EΓ × Γ C * (Γ) from equation (2).…”

“…, are defined from the isometries induced from inclusion of the relevant sets. In particular, the maps f and g induces partial isometries 2…”

Applying geometricK-cycles to fractional indices

“…Another geometric model, one that fits well also with non‐torsion twists, can be found in . A number of constructions related to those considered in this paper are discussed in . While there is some overlap between our results and those found in , we have concentrated more on specific applications while they concentrate more on the general theory.…”

“…A number of constructions related to those considered in this paper are discussed in . While there is some overlap between our results and those found in , we have concentrated more on specific applications while they concentrate more on the general theory. As such, these papers are, in fact, rather complementary.…”

“…It is at this point not clear if projective K ‐homology is the correct model for non‐torsion twists, but we include its definition due to the interesting examples. Another geometric model, one that fits well also with non‐torsion twists, can be found in . A number of constructions related to those considered in this paper are discussed in .…”