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The topology of the external activity complex of a matroid. (English) Zbl 1344.05036
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.

MSC:
05B35 Combinatorial aspects of matroids and geometric lattices
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
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