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a detailed proof of the existence-theorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997).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 * ðC r ðGÞÞ; for the index classes associated to 1-parameter family of Dirac operators on a G-covering with boundary; this formula involves a noncommutative spectral flow for the boundary family. Next, we establish an additivity result, in K * ðC r ðGÞÞ; for the index class defined by a Dirac-type operator associated to a closed manifold M and a map r : M-BG when we assume that M is the union along a hypersurface F of two manifolds with boundary M 1⁄4 Mþ ,F M : Finally, we prove a defect formula for the signature-index classes of two cut-and-paste equivalent pairs ðM1; r1 : M1-BGÞ and ðM2; r2 : M2-BGÞ; where M1 1⁄4 Mþ ,ðF ;f1Þ M ; M2 1⁄4 Mþ ,ðF ;f2Þ M and fjADiffðFÞ: The formula involves the noncommutative spectral flow of a suitable 1parameter 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. r 2003 Elsevier Science (USA). All rights reserved.