Dirac index classes and the noncommutative spectral flow
Eric Leichtnam and Paolo Piazza
We present a detailed proof of the existence-theorem for noncommutative spectral sections, see Wu [32]. 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 $K_*(C^*_r(\Gamma))$, 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 $r:M\rightarrow B\Gamma$ when we assume that $M$ is the union along a hypersurface $F$ of two manifolds with boundary: $M=M_+ \cup_F M_-$. Finally, we prove a defect formula for the signature-index classes of two cut-and-paste equivalent pairs $(M_1,r_1:M_1\rightarrow B\Gamma)$ and $(M_2,r_2:M_2\rightarrow B\Gamma)$, with $$M_1=M_+\cup_{(F,\phi_1)}M_-$ and $M_2=M_+\cup_{(F,\phi_2)}M_-.$$ 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.