The common basis complex and the partial decomposition poset
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arXiv
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| Format: | Preprint |
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2024
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| author | Brück, Benjamin Piterman, Kevin I. Welker, Volkmar |
| author_facet | Brück, Benjamin Piterman, Kevin I. Welker, Volkmar |
| contents | For a finite-dimensional vector space $V$, the common basis complex of $V$ is the simplicial complex whose vertices are the proper non-zero subspaces of $V$, and $σ$ is a simplex if and only if there exists a basis $B$ of $V$ that contains a basis of $S$ for all $S\in σ$. This complex was introduced by Rognes in 1992 in connection with stable buildings. In this article, we prove that the common basis complex is homotopy equivalent to the proper part of the poset of partial direct sum decompositions of $V$. Moreover, we establish this result in a more general combinatorial context, including the case of free groups, matroids, vector spaces with non-degenerate sesquilinear forms, and free modules over commutative Hermite rings, such as local rings or Dedekind domains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_10484 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The common basis complex and the partial decomposition poset Brück, Benjamin Piterman, Kevin I. Welker, Volkmar Combinatorics Algebraic Topology Group Theory K-Theory and Homology Primary 05E45, Secondary 29D50, 20E42, 55U10, 57M07 For a finite-dimensional vector space $V$, the common basis complex of $V$ is the simplicial complex whose vertices are the proper non-zero subspaces of $V$, and $σ$ is a simplex if and only if there exists a basis $B$ of $V$ that contains a basis of $S$ for all $S\in σ$. This complex was introduced by Rognes in 1992 in connection with stable buildings. In this article, we prove that the common basis complex is homotopy equivalent to the proper part of the poset of partial direct sum decompositions of $V$. Moreover, we establish this result in a more general combinatorial context, including the case of free groups, matroids, vector spaces with non-degenerate sesquilinear forms, and free modules over commutative Hermite rings, such as local rings or Dedekind domains. |
| title | The common basis complex and the partial decomposition poset |
| topic | Combinatorics Algebraic Topology Group Theory K-Theory and Homology Primary 05E45, Secondary 29D50, 20E42, 55U10, 57M07 |
| url | https://arxiv.org/abs/2402.10484 |