Combination of open covers with π1$\pi _1$‐constraints

Abstract Let be a group and let be a family of subgroups of . The generalised Lusternik–Schnirelmann category is the minimal cardinality of covers of by open subsets with fundamental group in . We prove a combination theorem for in terms of the stabilisers of contractible ‐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.