# Proper equivariant stable homotopy theory

## TL;DR

This paper develops a comprehensive framework for proper equivariant stable homotopy theory for Lie groups, introducing a model based on orthogonal G-spectra that captures equivariant cohomology theories with transfers and RO(G)-grading.

## Contribution

It constructs a model category for genuine proper G-equivariant stable homotopy theory using orthogonal G-spectra, incorporating transfers, Wirthmüller isomorphisms, and an RO(G)-grading.

## Key findings

- The category of orthogonal G-spectra models proper G-equivariant stable homotopy theory.
- Equivariant cohomology theories depend only on proper G-homotopy type, tested by fixed points.
- For discrete groups, the theory relates to finiteness properties and classical cohomology theories.

## Abstract

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective `proper' alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective `genuine' indicates that the theory comes with appropriate transfers and Wirthm\"uller isomorphisms, and the resulting equivariant cohomology theories support the analog of an $RO(G)$-grading. Our model for genuine proper $G$-equivariant stable homotopy theory is the category of orthogonal $G$-spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of $G$. This class of $\pi_*$-isomorphisms is part of a symmetric monoidal stable model structure and the associated tensor triangulated homotopy category is compactly generated. Every orthogonal $G$-spectrum represents an equivariant cohomology theory on the category of $G$-spaces, depending only on the `proper $G$-homotopy type', tested by fixed points under all compact subgroups. An important special case are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties, in the sense of geometric group theory; for example, the $G$-sphere spectrum is a compact object in the equivariant homotopy category if the universal space for proper $G$-actions has a finite $G$-CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper $G$-CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by $G$-vector bundles. Via this description, we can identify the previously defined $G$-cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal $G$-spectra.