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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Simplicial shellable spheres via combinatorial blowups


Authors: Sonja Lj. Cukic and Emanuele Delucchi
Journal: Proc. Amer. Math. Soc. 135 (2007), 2403-2414
MSC (2000): Primary 06A07, 55U10, 52B22
Published electronically: April 10, 2007
MathSciNet review: 2302561
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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: http://dx.doi.org/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
Published electronically: 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
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.