Multiple Cylinder of Relations for Finite Spaces and Nerve Theorem for Strong-Good Cover
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Das, Ponaki Mawiong, Sainkupar Marwein |
| author_facet | Das, Ponaki Mawiong, Sainkupar Marwein |
| contents | In this paper, we develop the concept of multiple cylinder of relations which is a generalization of the relation cylinder, extending the multiple non-Hausdorff mapping cylinder to sequences of finite T0-spaces linked by a series of relations. This construction is important in capturing complex homotopical structures across chains of finite spaces and, when the relations are induced by maps, it serves as a third space that collapses to two distinct finite spaces. Additionally, we introduce the concept of a strong-good cover for simplicial complexes and finite spaces, char acterized by collapsible (rather than merely contractible) intersections. This leads to a strengthened version of the Nerve Theorem, which we develop for simplicial complexes as well as for finite spaces with strong-good covers, demonstrating that these complexes and spaces and their associated nerves maintain the same simple homotopy type, thereby refining classical results for finite simplicial complexes and finite topological structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09281 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiple Cylinder of Relations for Finite Spaces and Nerve Theorem for Strong-Good Cover Das, Ponaki Mawiong, Sainkupar Marwein Algebraic Topology Combinatorics 06A99, 05E45, 57Q10 In this paper, we develop the concept of multiple cylinder of relations which is a generalization of the relation cylinder, extending the multiple non-Hausdorff mapping cylinder to sequences of finite T0-spaces linked by a series of relations. This construction is important in capturing complex homotopical structures across chains of finite spaces and, when the relations are induced by maps, it serves as a third space that collapses to two distinct finite spaces. Additionally, we introduce the concept of a strong-good cover for simplicial complexes and finite spaces, char acterized by collapsible (rather than merely contractible) intersections. This leads to a strengthened version of the Nerve Theorem, which we develop for simplicial complexes as well as for finite spaces with strong-good covers, demonstrating that these complexes and spaces and their associated nerves maintain the same simple homotopy type, thereby refining classical results for finite simplicial complexes and finite topological structures. |
| title | Multiple Cylinder of Relations for Finite Spaces and Nerve Theorem for Strong-Good Cover |
| topic | Algebraic Topology Combinatorics 06A99, 05E45, 57Q10 |
| url | https://arxiv.org/abs/2411.09281 |