Spheres arising from multicomplexes

Satoshi Murai*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex δ on the vertex set V with δ≢2V, the deleted join of δ with its Alexander dual δ∨ 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.

Original languageEnglish
Pages (from-to)2167-2184
Number of pages18
JournalJournal of Combinatorial Theory. Series A
Issue number8
Publication statusPublished - 2011 Nov
Externally publishedYes


  • Alexander duality
  • Bier spheres
  • Edge decomposability
  • Polarization
  • Shellability

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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