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Nonlinearity, Proper Actions and Equivariant Stable Cohomotopy

Noe Barcenas Torres,
published 2011-03-21

In this article we extend the classical de nitions of equivariant cohomotopy theory to the setting of proper actions of Lie groups. The main ideas we use come on the one hand from methods originally developed in the analysis of nonlinear di erential equations, mainly in connection with Leray-Schauder theory, and on the other hand from developments of equivariant K-Theory needed in noncommutative geometry, mainly done by N.C. Phillips. We prove the correspondence with a previous construction of W. Luck by generalizing an index theorem which has found useful applications in Partial Di erential Equations. As an illustration of these methods, we introduce a Burnside ring de ned in analytical terms. With this de nition, we extend a weak version of the Segal Conjecture to a certain family of Lie groups and generalize an invariant in Gauge Theory, due to Bauer and Furuta.

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