An index theorem for end-periodic operators

TL;DR

This paper extends the Atiyah-Patodi-Singer index theorem to manifolds with periodic ends, introducing a new periodic eta-invariant and applying it to moduli spaces of metrics with positive scalar curvature.

Contribution

It develops an index theorem for end-periodic operators, generalizing classical results to periodic end manifolds and introducing a novel periodic eta-invariant.

Findings

Established a periodic eta-invariant matching the classical eta-invariant in cylindrical cases

Extended index theory to manifolds with periodic ends

Applied the invariant to moduli spaces of positive scalar curvature metrics

Abstract

We extend the Atiyah, Patodi, and Singer index theorem for first order differential operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. This theorem provides a natural complement to Taubes' Fredholm theory for general end-periodic operators. Our index theorem is expressed in terms of a new periodic eta-invariant that equals the Atiyah-Patodi-Singer eta-invariant in the cylindrical setting. We apply this periodic eta-invariant to the study of moduli spaces of Riemannian metrics of positive scalar curvature.