Kervaire semi-characteristics in KK-theory and an Atiyah type vanishing theorem

TL;DR

This paper investigates Kervaire semi-characteristics on certain noncompact manifolds with group actions, establishing their equivalence through analytic and topological methods and proving a vanishing theorem akin to Atiyah's.

Contribution

It introduces a new framework connecting analytic and topological interpretations of Kervaire semi-characteristics on noncompact manifolds with group actions, leading to an Atiyah type vanishing result.

Findings

Analytic and topological interpretations of Kervaire semi-characteristics coincide.

Established an Atiyah type vanishing theorem for these semi-characteristics.

Extended the Hodge theorem to noncompact manifolds with proper cocompact group actions.

Abstract

On (4n + 1)-dimensional (noncompact) manifolds admitting proper cocompact Lie group actions, we explore the analytic and topological sides of Kervaire semi-characteristics. The analytic side puts together two interpretations, one via assembly maps, and the other via dimensions of kernels. The topological side is ensured by the proper cocompact version of the Hodge theorem. The two sides coincide and admit an Atiyah type vanishing theorem.