Shellability and homology of $q$-complexes and $q$-matroids

TL;DR

This paper introduces the concept of shellability for $q$-complexes, explores their homological properties, and demonstrates shellability for complexes derived from $q$-matroids, providing explicit homology calculations for uniform cases.

Contribution

It defines shellability for $q$-complexes, proves shellability for complexes from $q$-matroids, and computes their homology, advancing the understanding of $q$-analogues in combinatorial topology.

Findings

$q$-complexes from $q$-matroids are shellable

Homology of uniform $q$-matroid complexes is explicitly determined

Partial results on homology of general shellable $q$-complexes

Abstract

We consider a $q$-analogue of abstract simplicial complexes, called $q$-complexes, and discuss the notion of shellability for such complexes. It is shown that $q$-complexes formed by independent subspaces of a $q$-matroid are shellable. Further, we explicitly determine the homology of $q$-complexes corresponding to uniform $q$-matroids. We also outline some partial results concerning the determination of homology of arbitrary shellable $q$-complexes..