Twisted ambidexterity in equivariant homotopy theory

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 Wirthmüller 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 equivariant spectra over orbispaces and proper genuine equivariant spectra.