Multi-Dimensional Cohomological Phenomena in the Multiparametric Models of Random Simplicial Complexes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914917308694528 |
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| author | Kogan, Jon V. |
| author_facet | Kogan, Jon V. |
| contents | In the past two decades, extensive research has been conducted on the (co)homology of various models of random simplicial complexes. So far, it has always been examined merely as a list of groups. This paper expands upon this by describing both the ring structure and the Steenrod-algebra structure of the cohomology of the multiparametric models. For the lower model, we prove that the ring structure is always a.a.s trivial, while, for certain parameters, the Steenrod-algebra a.a.s acts non-trivially. This reveals that complex multi-dimensional topological structures appear as subcomplexes of this model. In contrast, we improve upon a result of Farber and Nowik, and assert that the cohomology of the upper multiparametric model is a.a.s concentrated in a single dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02573 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multi-Dimensional Cohomological Phenomena in the Multiparametric Models of Random Simplicial Complexes Kogan, Jon V. Algebraic Topology Combinatorics Probability 60D05 (Primary) 60C05, 55U10, 05D40, 05E45, 05C80, 55S10, 05C65 (Secondary) In the past two decades, extensive research has been conducted on the (co)homology of various models of random simplicial complexes. So far, it has always been examined merely as a list of groups. This paper expands upon this by describing both the ring structure and the Steenrod-algebra structure of the cohomology of the multiparametric models. For the lower model, we prove that the ring structure is always a.a.s trivial, while, for certain parameters, the Steenrod-algebra a.a.s acts non-trivially. This reveals that complex multi-dimensional topological structures appear as subcomplexes of this model. In contrast, we improve upon a result of Farber and Nowik, and assert that the cohomology of the upper multiparametric model is a.a.s concentrated in a single dimension. |
| title | Multi-Dimensional Cohomological Phenomena in the Multiparametric Models of Random Simplicial Complexes |
| topic | Algebraic Topology Combinatorics Probability 60D05 (Primary) 60C05, 55U10, 05D40, 05E45, 05C80, 55S10, 05C65 (Secondary) |
| url | https://arxiv.org/abs/2402.02573 |