2024/10/25 by Kai-Uwe Bux, Bux, Kai-Uwe, Elisa Hartmann +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GR #math.GT #msc:22D05 #msc:22F05
paper · pdf · doi:10.48550/arxiv.2410.19501
published as Bux, KU., Hartmann, E. & Quintanilha, J.P. Geometric Invariants of Locally Compact Groups: The Homotopical Perspective. Transformation Groups (2026) · 62 pages, 5 figures; version accepted for publication
arxiv created 2026/08/03 · arxiv updated 2026/08/04
We extend the classical theory of homotopical Σ-sets Σn developed by Bieri, Neumann, Renz and Strebel for abstract groups, to Σ-sets Σtopn for locally compact Hausdorff groups. Given such a group G, our Σtopn(G) are sets of continuous homomorphisms G → ℝ ("characters"). They match the classical Σ-sets Σn(G) if G is discrete, and refine the homotopical compactness properties \mathrm Cn of Abels and Tiemeyer. Moreover, our theory recovers the definition of Σtop1 and Σtop2 proposed by Kochloukova. Besides presenting various characterizations of Σtopn (particularly for n∈ \1,2\), we show that characters in Σtopn(G) are also in Σtopn(H) if H≤ G is a closed cocompact subgroup, and we generalize several classical results. Namely, we prove that the set of nonzero elements of Σtopn(G) is open, we prove that characters in a group of type \mathrm Cn that do not vanish on the center always lie in Σtopn(G), and we relate the Σ-sets of a group with those of its quotients by closed subgroups of type \mathrm Cn. Lastly, we describe how Σtopn(G) governs whether a closed normal subgroup with abelian quotient is of type \mathrm Cn, generalizing one of the highlights of the classical theory.