Geometric invariants of locally compact groups: the homological perspective

2024/11/20 by Kai-Uwe Bux, Bux, Kai-Uwe, Elisa Hartmann +3 · 1 citation
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2411.13272

openalex publication_date 2024/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we develop the homological version of Σ-theory for locally compact Hausdorff groups, leaving the homotopical version for another paper. Both versions are connected by a Hurewicz-like theorem. They can be thought of as directional versions of type CP_m and type C_m, respectively. And classical Σ-theory is recovered if we equip an abstract group with the discrete topology. This paper provides criteria for type CP_m and homological locally compact Σ^m. Given a short exact sequence with kernel of type CP_m, we can derive Σ^m of the extension on the sphere that vanishes on the kernel from the quotient and likewise. Given a short exact sequence with abelian quotient, Σ-theory on the extension can tell if the kernel is of type CP_m.

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