2024/05/27 by Kaif Hilman, Hilman, Kaif, Dominik Kirstein +3 · 2 citations
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Algebraic Topology (math.AT) #Duality (order theory) #Equivariant map #FOS: Mathematics #Fixed Point Theorems Analysis #Fixed point #Geometric Topology (math.GT) #Mathematical analysis #Mathematics #Optimization and Variational Analysis #Poincaré conjecture #Poincaré duality #Pure mathematics
paper · pdf · doi:10.48550/arxiv.2405.17641
openalex publication_date 2024/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we introduce and develop the notion of parametrised Poincaré 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 Poincaré 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.