The topology of the external activity complex of a matroid

TL;DR

This paper proves that the external activity complex of a matroid is shellable, relates its structure to linear extensions of certain orders, and characterizes its topological type based on minors.

Contribution

It establishes shellability of the external activity complex and links its combinatorial and topological properties to matroid minors and linear extensions.

Findings

External activity complex is shellable.

Shelling can be obtained from linear extensions of specific orders.

Topological type depends on the presence of a $U_{3,1}$ minor.

Abstract

We prove that the external activity complex $\textrm{Act}_<(M)$ of a matroid is shellable. In fact, we show that every linear extension of LasVergnas's external/internal order $<_{ext/int}$ on $M$ provides a shelling of $\textrm{Act}_<(M)$. We also show that every linear extension of LasVergnas's internal order $<_{int}$ on $M$ provides a shelling of the independence complex $IN(M)$. As a corollary, $\textrm{Act}_<(M)$ and $M$ have the same $h$-vector. We prove that, after removing its cone points, the external activity complex is contractible if $M$ contains $U_{3,1}$ as a minor, and a sphere otherwise.