Dirac index classes and the noncommutative spectral flow

Abstract

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a detailed proof of the existence-theorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997).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 ∗(Γ)) , for the index classes associated to 1-parameter family of Dirac operators on a Γ-covering with boundary; this formula involves a noncommutative spectral flow for the boundary family. Next, we establish an additivity result, in K ∗(C r ∗(Γ)) , for the index class defined by a Dirac-type operator associated to a closed manifold M and a map r: M→ BΓ when we assume that M is the union along a hypersurface F of two manifolds with boundary M=M + ∪ F 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→ BΓ) and ( M 2, r 2: M 2→ BΓ), where M 1=M + ∪ (F,φ 1) M −, M 2=M + ∪ (F,φ 2) M − and φ j ∈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.