Simplicial complexes Alexander dual to boundaries of polytopes

TL;DR

This paper explores Gale diagrams and Alexander duality to characterize simplicial complexes related to polytopes, establishing a combinatorial method to determine polytope properties like the Buchstaber invariant and constructing polytopes with specific invariants.

Contribution

It introduces a combinatorial approach using Gale diagrams to describe Alexander dual complexes and characterizes polytopes with Buchstaber invariant 1, also providing a construction method for polytopes with higher invariants.

Findings

Buchstaber invariant s(P)=1 iff P is a pyramid

A procedure to construct polytopes with s_R(P)>k

Application of Gale duality to Betti numbers and f-vectors

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.