Balanced sets and homotopy invariants of covers
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915562147282944 |
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| author | Bludov, Mikhail V. |
| author_facet | Bludov, Mikhail V. |
| contents | In this paper, we study a construction of homotopy invariants of open or closed covers, where the homotopy class is defined relative to a pair $(V,r)$, with $V$ a finite set of points in $\mathbb{R}^d$ and $r$ a point in the interior of their convex hull. We show that the simplicial complex of non-balanced subsets associated with $(V,r)$ has the homotopy type of a sphere, and use this to develop a theory of homotopy invariants of covers relative to balanced sets. A key result is that the homotopy class of a cover depends only, up to an involution, on the balanced-equivalence class of $(V,r)$. As applications, we obtain extension theorems for covers in this setting and derive the KKMS lemma, its analogues, and related combinatorial fixed-point results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05799 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Balanced sets and homotopy invariants of covers Bludov, Mikhail V. Combinatorics In this paper, we study a construction of homotopy invariants of open or closed covers, where the homotopy class is defined relative to a pair $(V,r)$, with $V$ a finite set of points in $\mathbb{R}^d$ and $r$ a point in the interior of their convex hull. We show that the simplicial complex of non-balanced subsets associated with $(V,r)$ has the homotopy type of a sphere, and use this to develop a theory of homotopy invariants of covers relative to balanced sets. A key result is that the homotopy class of a cover depends only, up to an involution, on the balanced-equivalence class of $(V,r)$. As applications, we obtain extension theorems for covers in this setting and derive the KKMS lemma, its analogues, and related combinatorial fixed-point results. |
| title | Balanced sets and homotopy invariants of covers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.05799 |