Parametrised Poincaré duality and equivariant fixed points methods

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.