Cobordism invariance of the index for realizations of elliptic operators revisited

TL;DR

This paper revisits classical cobordism invariance proofs for elliptic operator indices and extends results to spectral flow vanishing for families of selfadjoint Fredholm realizations, enhancing understanding of index behavior under cobordisms.

Contribution

It combines historical and recent methods to prove new vanishing results for spectral flow in families of elliptic operators induced on boundaries.

Findings

Proves cobordism invariance of elliptic operator indices using revisited arguments.

Establishes vanishing results for spectral flow in certain families of operators.

Links index behavior under cobordisms to boundary-induced operator families.

Abstract

We revisit an argument due to Lesch (Topology 32 (1993), no. 3, 611-623) for proving the cobordism invariance of the index of Dirac operators on even-dimensional closed manifolds and combine this with recent work by the author (New York J. Math. 28 (2022), 705-772) to show vanishing results for the spectral flow for families of selfadjoint Fredholm realizations of elliptic operators in case the family is induced on the boundary by an elliptic operator on a compact space. This work is motivated by studying the behavior of the index of realizations of elliptic operators under cobordisms of statified manifolds.