Spheres arising from multicomplexes

TL;DR

This paper extends Bier's construction of simplicial spheres to multicomplexes, revealing a new class of shellable, edge decomposable spheres with algebraic connections, not realizable as polytopes.

Contribution

It generalizes Bier's construction to multicomplexes and explores their combinatorial and algebraic properties, introducing new non-polytopal shellable spheres.

Findings

All constructed spheres are shellable and edge decomposable.

The spheres are related to polarizations and Alexander duality in monomial ideals.

These spheres are not realizable as polytopes.

Abstract

In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex $\Delta$ on the vertex set $V$ with $\Delta \ne 2^V$, the deleted join of $\Delta$ with its Alexander dual $\Delta^\vee$ is a combinatorial sphere. In this paper, we extend Bier's construction to multicomplexes, and study their combinatorial and algebraic properties. We show that all these spheres are shellable and edge decomposable, which yields a new class of many shellable edge decomposable spheres that are not realizable as polytopes. It is also shown that these spheres are related to polarizations and Alexander duality for monomial ideals which appear in commutative algebra theory.