Spheres arising from multicomplexes

Details Spheres arising from multicomplexes Spheres arising from multicomplexes

2010-02-05

arxiv.org/abs/1002.1211v3

Mathematics

Combinatorics

20 pages. Improve presentation. To appear in Journal of Combinatorial Theory, Series A

Scientific paper

10.1016/j.jcta.2011.04.015

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.

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