The topology of the external activity complex of a matroid

Abstract

We prove that the external activity complex ${\text{Act}}_{<}(M)$ 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 ${\text{Act}}_{<}(M)$. 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, ${\text{Act}}_{<}(M)$ 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.

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