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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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1615-7168
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Volume 14, Issue 3

Issues

Non-existence of tight neighborly triangulated manifolds with β1 = 2

N. Singh
Published Online: 2014-07-08 | DOI: https://doi.org/10.1515/advgeom-2014-0005

Abstract

All triangulated d-manifolds satisfy the inequality for d ≥ 3. A triangulated d-manifold is called tight neighborly if it attains equality in this bound. For each d ≥ 3, a (2d+3)-vertex tight neighborly triangulation of the Sd-1-bundle over S1 with β1 = 1 was constructed by Kühnel in 1986. In this paper, it is shown that there does not exist a tight neighborly triangulated manifold with β1 = 2. In other words, there is no tight neighborly triangulation of (Sd-1x S1)#2 or (Sd-1x̲ S1)#2 for d ≥ 3. A short proof of the uniqueness of Kühnel’s complexes for d ≥ 4 under the assumption β1≠ 0 is also presented.

Keywords: Stacked sphere; triangulated manifolds; tight neighborly triangulation

About the article

Published Online: 2014-07-08

Published in Print: 2014-07-01


Citation Information: Advances in Geometry, Volume 14, Issue 3, Pages 561–569, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2014-0005.

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[1]
Basudeb Datta and Nitin Singh
Journal of Combinatorial Theory, Series A, 2013, Volume 120, Number 8, Page 2148
[2]
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Proceedings - Mathematical Sciences, 2015, Volume 125, Number 1, Page 79

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