Combination of open covers with $π_1$-constraints
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916724782137344 |
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| author | Capovilla, Pietro Li, Kevin Loeh, Clara |
| author_facet | Capovilla, Pietro Li, Kevin Loeh, Clara |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04292 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Combination of open covers with $π_1$-constraints Capovilla, Pietro Li, Kevin Loeh, Clara Algebraic Topology Group Theory 55M30, 57M07 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. |
| title | Combination of open covers with $π_1$-constraints |
| topic | Algebraic Topology Group Theory 55M30, 57M07 |
| url | https://arxiv.org/abs/2505.04292 |