The complex of partial bases of a free group

TL;DR

This paper proves that the complex formed by partial bases of a free group is homotopy equivalent to a wedge of spheres and has Cohen-Macaulay properties, advancing understanding of free group structures.

Contribution

It establishes the homotopy type and Cohen-Macaulay property of the complex of partial bases in free groups, a novel topological insight.

Findings

Complex is homotopy equivalent to a wedge of (n-1)-spheres

Complex is Cohen-Macaulay

Provides new topological understanding of free group bases

Abstract

We prove that the simplicial complex whose simplices are the nonempty partial bases of $\mathbb{F}_n$ is homotopy equivalent to a wedge of $(n-1)$-spheres. Moreover, we show that it is Cohen-Macaulay.