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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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