Spin-structures and proper group actions

TL;DR

This paper extends a classical vanishing theorem to broader settings involving noncompact groups and manifolds, providing new insights into Spin-geometry and group actions.

Contribution

It generalizes Atiyah and Hirzebruch's vanishing theorem to proper, possibly noncompact group actions on noncompact Spin-manifolds, expanding its applicability.

Findings

Vanishing results for $\hat A$-type genera under broader group actions

Extension of classical theorems to noncompact group actions

New conditions for Spin-manifolds with proper group actions

Abstract

We generalise Atiyah and Hirzebruch's vanishing theorem for actions by compact groups on compact Spin-manifolds to possibly noncompact groups acting properly and cocompactly on possibly noncompact Spin-manifolds. As corollaries, we obtain some vanishing results for $\hat A$-type genera.