Dirac index classes and the noncommutative spectral flow

We present a detailed proof of the existence-theorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997). We apply this result to various index-theoretic situations, extending to the noncommutative context results of Booss-Wojciechowski, Melrose-Piazza and Dai-Zhang. In particular, we prove a variational formula, in K-*(C-r(*)(Gamma)), for the index classes associated to 1-parameter family of Dirac operators on a Gamma-covering with boundary; this formula involves a noncommutative spectral flow for the boundary family. Next, we establish an additivity result, in K-* (C-r(*)(Gamma)), for the index class defined by a Dirac-type operator associated to a closed manifold M and a map r : M --> BGamma when we assume that M is the union along a hypersurface F of two manifolds with boundary M = M+ boolean ORF M-. Finally, we prove a defect formula for the signature-index classes of two cut-and-paste equivalent pairs (M-1, r(1) : M-1 --> BGamma) and (M-2, r(2) : M-2 --> BGamma), where M-1 = M+ boolean OR (F, phi(1)) M- M-2 = M+ boolean OR (F, phi(2)) and phi(1)is an element of Diff (F). The formula involves the noncommutative spectral flow of a suitable 1-parameter family of twisted signature operators on F. We give applications to the problem of cut-and-paste invariance of Novikov's higher signatures on closed oriented manifolds. (C) 2003 Elsevier Science (USA). All rights reserved.