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Simplicial shellable spheres via combinatorial blowups

Author(s): Sonja Lj. Cukic; Emanuele Delucchi
Journal: Proc. Amer. Math. Soc. 135 (2007), 2403-2414.
MSC (2000): Primary 06A07, 55U10, 52B22
Posted: April 10, 2007
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Abstract: The construction of the Bier sphere $ \textrm{Bier}(K)$ for a simplicial complex $ K$ is due to Bier (1992). Björner, Paffenholz, Sjöstrand and Ziegler (2005) generalize this construction to obtain a Bier poset $ \textrm{Bier}(P,I)$ from any bounded poset $ P$ and any proper ideal $ I\subseteq P$. They show shellability of $ \textrm{Bier}(P,I)$ for the case $ P=B_n$, the boolean lattice, and thereby obtain `many shellable spheres' in the sense of Kalai (1988).

We put the Bier construction into the general framework of the theory of nested set complexes of Feichtner and Kozlov (2004). We obtain `more shellable spheres' by proving the general statement that combinatorial blowups, hence stellar subdivisions, preserve shellability.

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Additional Information:

Sonja Lj. Cukic
Affiliation: Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland
Email: sonja@math.binghamton.edu

Emanuele Delucchi
Affiliation: Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
Email: delucchi@mail.dm.unipi.it

DOI: 10.1090/S0002-9939-07-08768-0
PII: S 0002-9939(07)08768-0
Keywords: Posets, lattices, shellability, combinatorial blowups, building sets, nested sets, simplicial shellable spheres, Bier posets, Bier lattices
Received by editor(s): February 2, 2006
Received by editor(s) in revised form: May 2, 2006
Posted: April 10, 2007
Additional Notes: Research partially supported by TH-Projekt 0-20268-05, and by the Swiss National Science Foundation, project PP002--106403/1
Communicated by: Paul Goerss
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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