The two definitions of the index difference

TL;DR

This paper proves that two different definitions of a secondary index invariant for positive scalar curvature metrics on spin manifolds are equivalent and extends this result to families of metrics, generalizing the spectral-flow-index theorem.

Contribution

It provides a detailed proof that homotopical and index problem definitions of the invariant coincide and generalizes the spectral-flow-index theorem to families of operators.

Findings

Both definitions of the invariant agree.

The result extends to families parametrized by compact spaces.

Generalization of the spectral-flow-index theorem.

Abstract

Given two metrics of positive scalar curvature metrics on a closed spin manifold, there is a secondary index invariant in real $K$-theory. There exist two definitions of this invariant, one of homotopical flavour, the other one defined by a index problem of Atiyah-Patodi-Singer type. We give a complete and detailed proof of the folklore result that both constructions yield the same answer. Moreover, we generalize this to the case of two families of positive scalar curvature metrics, parametrized by a compact space. In essence, we prove a generalization of the classical "spectral-flow-index theorem" to the case of families of real operators.