A mod 2 index theorem for pin$^-$ manifolds

TL;DR

This paper proves a mod 2 index theorem for real vector bundles over non-orientable pin$^-$ manifolds, extending Atiyan and Singer's work to a broader class of manifolds using $KO$-theory and Dirac operators.

Contribution

It introduces a new mod 2 index theorem for pin$^-$ manifolds, generalizing previous results to non-orientable cases with a $KO$-theoretic approach.

Findings

Established a mod 2 index theorem for pin$^-$ manifolds.

Connected the analytic index with the topological index via $KO$-theory.

Extended Atiyan and Singer's theorem to non-orientable manifolds.

Abstract

We establish a mod 2 index theorem for real vector bundles over 8k+2 dimensional compact pin$^-$ manifolds. The analytic index is the reduced $\eta$ invariant of (twisted) Dirac operators and the topological index is defined through $KO$-theory. Our main result extends the mod 2 index theorem of Atiyan and Singer to non-orientable manifolds.