Notes on the topology of independence structures
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913996307693568 |
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| author | Piterman, Kevin Ivan Welker, Volkmar |
| author_facet | Piterman, Kevin Ivan Welker, Volkmar |
| contents | Following Welsh, a pre-independence space (pi-space) is a set $M$ together with a non-empty collection $I(M)$ of subsets of $M$, called independent sets, which is closed under taking subsets, and finite independent sets satisfy the exchange property from matroid theory.
We show that $I(M)$, viewed as a poset, is contractible if it is infinite-dimensional, and Cohen-Macaulay otherwise. Moreover, the proper part of the associated poset of flats is also contractible in the infinite-dimensional case, and Cohen-Macaulay otherwise.
These results generalize those for independence complexes and geometric lattices of (finite) matroids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12971 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Notes on the topology of independence structures Piterman, Kevin Ivan Welker, Volkmar Combinatorics Algebraic Topology Following Welsh, a pre-independence space (pi-space) is a set $M$ together with a non-empty collection $I(M)$ of subsets of $M$, called independent sets, which is closed under taking subsets, and finite independent sets satisfy the exchange property from matroid theory. We show that $I(M)$, viewed as a poset, is contractible if it is infinite-dimensional, and Cohen-Macaulay otherwise. Moreover, the proper part of the associated poset of flats is also contractible in the infinite-dimensional case, and Cohen-Macaulay otherwise. These results generalize those for independence complexes and geometric lattices of (finite) matroids. |
| title | Notes on the topology of independence structures |
| topic | Combinatorics Algebraic Topology |
| url | https://arxiv.org/abs/2508.12971 |