Combination of open covers with $\pi_1$-constraints

TL;DR

This paper develops a combination theorem for a generalized Lusternik--Schnirelmann category involving $ ext{pi}_1$-constraints, with applications to manifold invariants and topological complexity.

Contribution

It introduces a new combination theorem for $ ext{cat}_ ext{F}(G)$ based on stabilisers of $G$-CW-complexes, extending previous results and applying to manifold and topological complexity problems.

Findings

Vanishing results for simplicial volume of manifold gluings

Upper bounds for Farber's topological complexity

Generalization of estimates for amalgamated products

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.