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arXiv · 1310.5487

Simplicial complexes Alexander dual to boundaries of polytopes

Anton Ayzenberg
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Abstract

In the paper we treat Gale diagrams in a combinatorial way. The interpretation allows to describe simplicial complexes which are Alexander dual to boundaries of simplicial polytopes and, more generally, to nerve-complexes of general polytopes. This technique and recent results of N.Yu.Erokhovets are combined to prove the following: Buchstaber invariant $s(P)$ of a convex polytope equals 1 if and only if $P$ is a pyramid. In general, we describe a procedure to construct polytopes with $s_R(P)>k$. The construction has purely combinatorial consequences. We also apply Gale duality to the study of bigraded Betti numbers and f-vectors of polytopes.

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Anton Ayzenberg. 2013-10-21. Simplicial complexes Alexander dual to boundaries of polytopes. https://arxiv.org/abs/1310.5487

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