Affiliation:
1. Institut für Mathematische Logik und Grundlagenforschung, Universität Münster , 48149 Münster, Germany
2. Fachbereich Mathematik und Informatik, Philipps-Universität Marburg , 35032 Marburg, Germany
Abstract
Abstract
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 $\sigma $ is a simplex if and only if there exists a basis $B$ of $V$ that contains a basis of $S$ for all $S\in \sigma $. 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.
Funder
Deutsche Forschungsgemeinschaft
Germany’s Excellence Strategy
Alexander von Humboldt Stiftung
Fonds voor Wetenschappelijk Onderzoek–Vlaanderen
Publisher
Oxford University Press (OUP)
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