Combination of open covers with π₁-constraints

2025/05/07 by Pietro Capovilla, Kevin Li, Capovilla, Pietro +3
Computer Science · Mathematics · #55M30 #57M07 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2505.04292

openalex publication_date 2025/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

Let~G be a group and let~F be a family of subgroups of~G. The generalised Lusternik--Schnirelmann category~cat_F(G) is the minimal cardinality of covers of~BG by open subsets with fundamental group in~F. We prove a combination theorem for~cat_F(G) in terms of the stabilisers of contractible G-CW-complexes. As applications for the amenable category, we obtain vanishing results for the simplicial volume of gluings of manifolds (along not necessarily amenable boundaries) and of cyclic branched coverings. Moreover, we deduce an upper bound for Farber's topological complexity, generalising an estimate for amalgamated products of Dranishnikov--Sadykov.

Citations

• On the (super)additivity of simplicial volume
• Amenable covers of right-angled Artin groups
• Bounded acyclicity and relative simplicial volume
• Amenable covers and integral foliated simplicial volume
• Minimal volume entropy and fiber growth
• Brown's Criterion and classifying spaces for families
• Higher topological complexity of aspherical spaces
• The topological complexity of the free product
• Bounded cohomology of discrete groups
• Isometric Embeddings in Bounded Cohomology
• Amenable covers, volume and L2-Betti numbers of aspherical manifolds
• Topological complexity of motion planning

Related

• On the Coarse Lusternik-Schnirelmann Category of Groups
• On the relative Lusternik-Schnirelmann category with respect to a closed 1-form
• Covers of groups definable in o-minimal structures
• Estimates of covering type and minimal triangulations based on category weight
• Group covers, o-minimality, and categoricity