The topology of the external activity complex of a matroid (Q311493)

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