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

 

On homology spheres with few minimal non-faces


Author: Lukas Katthän
Journal: Proc. Amer. Math. Soc. 140 (2012), 2489-2500
MSC (2010): Primary 52B05, 05E45; Secondary 13F55
Published electronically: November 9, 2011
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Abstract: Let $ \Delta $ be a $ (d-1)$-dimensional homology sphere on $ n$ vertices with $ m$ minimal non-faces. We consider the invariant $ \alpha (\Delta ) = m - (n-d)$ and prove that for a given value of $ \alpha $, there are only finitely many homology spheres that cannot be obtained through one-point suspension and suspension from another. Moreover, we describe all homology spheres with $ \alpha (\Delta )$ up to four and, as a corollary, all homology spheres with up to eight minimal non-faces. To prove these results we consider the lcm-lattice and the nerve of the minimal non-faces of $ \Delta $. Also, we give a short classification of all homology spheres with $ n-d \leq 3$.


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Additional Information

Lukas Katthän
Affiliation: Fachbereich Mathematik und Informatik, Philipps-Universität, 35032 Marburg, Germany
Email: katthaen@mathematik.uni-marburg.de

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11095-5
PII: S 0002-9939(2011)11095-5
Received by editor(s): February 1, 2011
Received by editor(s) in revised form: February 23, 2011
Published electronically: November 9, 2011
Additional Notes: This work was partially supported by the DAAD and the DFG
Communicated by: Irena Peeva
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.