Twisted K-theory with coefficients in a C*-algebra and obstructions against positive scalar curvature metrics

Abstract

We introduce the notion of twisted Hilbert A-module bundles to construct a geometric realization of twisted K-theory with coefficients in a C*-algebra A. Like ordinary bundles of Hilbert A-modules provide a generalization of vector bundles, twisted Hilbert A-module bundles extend the concept of modules over bundle gerbes. Their corresponding Grothendieck group is isomorphic to K_0(C(M,S)), where S denotes a bundle of C*-algebras, such that its structure group reduces to PU(A) = U(A)/U(1). In case S is a bundle of matrix algebras the above description boils down to ordinary twisted K-theory, therefore K_0(C(M,S)) can be understood as twisted K-theory with coefficients in A.If the structure group of S reduces to a Lie group G (an assumption, which is satisfied in the geometric applications we consider), classical notions from differential geometry and index theory have counterparts in the twisted theory. In particular, we define connections on twisted bundles and a Chern character, which takes values in the cohomology with coefficients in (K_0(A) \otimes R). By countertwisting with bundles of opposite twist we prove generalizations of the index theorem of Mishchenko and Fomenko. Since flat countertwists play a central role in our applications, we give a classification of flat modules over bundle gerbes via their (projective) holonomy. Apart from that, we consider transversally elliptic pseudodifferential operators on twisted Hilbert A-module bundles and define a distribution-valued index for the case, that A admits a trace.Algebras of the form C(M,S) appear quite naturally in index problems on manifolds without a K-orientation. In particular, the twisted K-homology of such manifolds still contains an element, which is induced by a Dirac-type operator and can be viewed as a replacement for the fundamental class. We prove that the Rosenberg index in the twisted case decomposes as a pairing of this class with a corresponding twisted Hilbert A-module bundle. This invariant is an index obstruction to the existence of positive scalar curvature metrics. It was proven by Hanke and Schick that enlargeable spin manifolds have a non-vanishing Rosenberg index. As an application of our theory we extend this result to arbitrary enlargeable manifolds.