Parametrised Poincar\'e duality and equivariant fixed points methods

TL;DR

This paper develops a parametrised Poincaré duality framework within higher category theory, generalising twisted ambidexterity, and applies it to equivariant topology to extend classical fixed point theorems.

Contribution

It introduces parametrised Poincaré duality in higher categories, generalises twisted ambidexterity, and applies these concepts to equivariant fixed point theory for Lie groups.

Findings

Established basechange results for coefficient categories

Developed a theory of equivariant Poincaré duality spaces

Generalised Atiyah-Bott and Conner-Floyd fixed point theorems

Abstract

In this article, we introduce and develop the notion of parametrised Poincar\'{e} duality in the formalism of parametrised higher category theory by Martini-Wolf, in part generalising Cnossen's theory of twisted ambidexterity to the nonpresentable setting. We prove several basechange results, allowing us to move between different coefficient categories and ambient topoi. We then specialise the general framework to yield a good theory of equivariant Poincar\'{e} duality spaces for compact Lie groups and apply our basechange results to obtain a suite of isotropy separation methods. Finally, we employ this theory to perform various categorical Smith-theoretic manoeuvres to prove, among other things, a generalisation of a theorem of Atiyah-Bott and Conner-Floyd on group actions with single fixed points.