Twisted ambidexterity in equivariant homotopy theory

TL;DR

This paper introduces twisted ambidexterity in parametrized $$-categories, generalizing key concepts in equivariant homotopy theory, and characterizes genuine $G$-spectra as a universal example satisfying these properties.

Contribution

It develops the theory of twisted ambidexterity in parametrized $$-categories and establishes the universal property of genuine $G$-spectra within this framework.

Findings

Genuine $G$-spectra form a universal example of twisted ambidexterity.

Extension of results to orbispectra and proper $G$-spectra for non-compact Lie groups.

Generalization of Wirthmüller isomorphisms and Costenoble-Waner duality.

Abstract

We develop the concept of twisted ambidexterity in a parametrized presentably symmetric monoidal $\infty$-category, which generalizes the notion of ambidexterity by Hopkins and Lurie and the Wirthm\"uller isomorphisms in equivariant stable homotopy theory, and is closely related to Costenoble-Waner duality. Our main result establishes the parametrized $\infty$-category of genuine $G$-spectra for a compact Lie group $G$ as the universal example of a presentably symmetric monoidal $\infty$-category parametrized over $G$-spaces which is both stable and satisfies twisted ambidexterity for compact $G$-spaces. We further extend this result to the settings of orbispectra and proper genuine $G$-spectra for a Lie group $G$ which is not necessarily compact.