Combination of open covers with $π_1$-constraints

TL;DR

The paper develops a general combination framework for the $\mathcal{F}$-category $\operatorname{cat}_\mathcal{F}(G)$ using contractible $G$-CW-complexes and equivariant pushouts, enabled by Lück's blow-up model of $EG\times X$. It provides two main bounds (a maximal bound and a summation bound) and applies them to graphs of groups, yielding vanishing results for simplicial volume in gluings and cyclic branched coverings, as well as an upper bound for Farber's topological complexity $\mathop{\mathrm{TC}}(G)$. The results cover cases with non-amenable boundaries and extend prior amalgamated-product estimates, connecting geometric group theory with manifold topology. Overall, the work offers a versatile toolkit for deriving volume vanishing and complexity bounds from group-theoretic decompositions via the $\mathcal{F}$-category framework.

Abstract

Let~$G$ be a group and let~$\mathcal{F}$ be a family of subgroups of~$G$. The generalised Lusternik--Schnirelmann category~$\operatorname{cat}_\mathcal{F}(G)$ is the minimal cardinality of covers of~$BG$ by open subsets with fundamental group in~$\mathcal{F}$. We prove a combination theorem for~$\operatorname{cat}_\mathcal{F}(G)$ 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.

Figures (1)

• Figure 1: The manifolds $W$ and $W\!\langle{3}\rangle$, schematically

Theorems & Definitions (41)

• Theorem 1.1: Brown82Geoghegan08
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6: Gromov82
• Definition 2.1
• Lemma 2.2
• Proposition 2.3: CLM22
• Lemma 2.4