The common basis complex and the partial decomposition poset
Abstract
For a finitedimensional vector space $V$, the common basis complex of $V$ is the simplicial complex whose vertices are the proper nonzero 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 nondegenerate sesquilinear forms, and free modules over commutative Hermite rings, such as local rings or Dedekind domains.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.10484
 arXiv:
 arXiv:2402.10484
 Bibcode:
 2024arXiv240210484B
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Topology;
 Mathematics  Group Theory;
 Mathematics  KTheory and Homology;
 Primary 05E45;
 Secondary 29D50;
 20E42;
 55U10;
 57M07
 EPrint:
 16 pages