DIRAC INDEX CLASSES AND THE NONCOMMUTATIVE SPECTRAL FLOW

ERIC LEICHTNAM PAOLO PIAZZA

Abstract. We present a detailed proof of the existence-theorem for noncommutative spectral sections, see [?]. We apply this result to various index-theoretic situations, extending to the noncommutative context results of Melrose- Piazza, Dai-Zhang and Booss-Wojciechowski. 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), with M_{1} = M_{+} ∪_{(F,ϕ_{1})} M_{−} and M_{2} = M_{+} ∪_{(F,ϕ_{2})} M_{−}. 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.