Nonlinearity, Proper Actions and Equivariant Stable Cohomotopy

TL;DR

This paper extends equivariant cohomotopy theory to proper Lie group actions, integrating nonlinear analysis and K-theory, and introduces new invariants with applications to the Segal Conjecture and Gauge Theory.

Contribution

It develops a new framework for equivariant cohomotopy for proper Lie group actions, combining analytical and algebraic methods, and introduces a Burnside ring with applications to conjectures and gauge invariants.

Findings

Constructed an index linking different equivariant cohomotopy models.

Extended a weak version of the Segal Conjecture to certain Lie groups.

Introduced an analytical Burnside ring related to gauge invariants.

Abstract

In this article we extend the classical definitions of equivariant cohomotopy theory to the setting of proper actions of Lie groups. We combine methods originally developed in the analysis of nonlinear differential equations, mainly in connection with Leray-Schauder theory, and on the other hand from developments of equivariant $K$-Theory by N.C. Phillips. We prove the correspondence with a previous construction of W. L\"uck by constructing an index. As an illustration of these methods, we introduce a Burnside ring defined in analytical terms. With this definition, we extend a weak version of the Segal Conjecture to a certain family of Lie groups and comment the relation to an invariant in Gauge Theory, due to Bauer and Furuta.