Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-2011-11095-5
Published electronically: November 9, 2011
MathSciNet review: 2898711
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • 1. D. Barnette, The triangulations of the $ 3$-sphere with up to $ 8$ vertices, Journal of Combinatorial Theory, Series A 14 (1973), no. 1, 37-52. MR 0312511 (47:1068)
  • 2. V. Gasharov, I. Peeva, and V. Welker, The LCM-lattice in monomial resolutions, Mathematical Research Letters 6 (1999), no. 5/6, 521-532. MR 1739211 (2001e:13018)
  • 3. T. Hibi, K. Kimura, and S. Murai, Betti numbers of chordal graphs and f-vectors of simplicial complexes, Journal of Algebra 323 (2010), no. 6, 1678-1689. MR 2588131 (2011b:13071)
  • 4. M. Joswig and F.H. Lutz, One-point suspensions and wreath products of polytopes and spheres, Journal of Combinatorial Theory, Series A 110 (2005), no. 2, 193-216. MR 2142174 (2006b:57033)
  • 5. Y. Kamoi, On Gorenstein monomial ideals of codimension three, Rocky Mt. J. Math. 25 (1995), no. 4, 1385-1393. MR 1371345 (97d:13009)
  • 6. P. Mani, Spheres with few vertices, Journal of Combinatorial Theory, Series A 13 (1972), no. 3, 346-352. MR 0317175 (47:5723)
  • 7. R.P. Stanley, Combinatorics and commutative algebra, Birkhäuser, 1996. MR 1453579 (98h:05001)
  • 8. G.M. Ziegler, Lectures on polytopes, Springer, 1995. MR 1311028 (96a:52011)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 52B05, 05E45, 13F55

Retrieve articles in all journals with MSC (2010): 52B05, 05E45, 13F55


Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-2011-11095-5
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.

American Mathematical Society