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a detailed proof of the existence-theorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997).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∗(Cr∗(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∗(Cr∗(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 (M1,r1:M1→BΓ) and (M2,r2:M2→BΓ), whereM1=M+ ∪(F,φ1) M−, M2=M+ ∪(F,φ2) M−and φj∈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 Novikovʹs higher signatures on closed oriented manifolds.