2024/12/02 by David Baron, Baron, David, Urshita Pal +9
Computer Science · Mathematics · #Amino acid #Biology #Business #Computer science #Genetics #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Mathematics #Political science #Pure mathematics #Representation (politics) #Stability (learning theory) #Topological and Geometric Data Analysis #math.AT #math.CO
paper · pdf · doi:10.48550/arxiv.2412.01128
openalex publication_date 2024/12/02 · openalex created_date 2024/12/05 · openalex updated_date 2026/07/29
In this paper, we study sequences of topological spaces called "vertical configuration spaces" of points in Euclidean space. We apply the theory of FIG-modules, and results of Bianchi-Kranhold, to show that their (co)homology groups are "representation stable" with respect to natural actions of wreath products Sk \wr Sn. In particular, we show that in each (co)homological degree, the (co)homology groups (viewed as Sk \wr Sn-representations) can be expressed as induced representations of a specific form. Consequently, the characters of their rational (co)homology groups, and the patterns of irreducible Sk \wr Sn-representation constituents of these groups, stabilize in a strong sense. In addition, we give a new proof of rational (co)homological stability for unordered vertical configuration spaces, with an improved stable range.