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