On Walkup's class K (d) and a minimal triangulation of (S-3 (sic) S-1)(#3)

Bagchi, Bhaskar and Datta, Basudeb (2011) On Walkup's class K (d) and a minimal triangulation of (S-3 (sic) S-1)(#3). In: Discrete Mathematics, 311 (12). pp. 989-995.

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Abstract

For d >= 2, Walkup's class K (d) consists of the d-dimensional simplicial complexes all whose vertex-links are stacked (d - 1)-spheres. Kalai showed that for d >= 4, all connected members of K (d) are obtained from stacked d-spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic chi satisfies f(1) >= 5f(0) - 15/2 chi, with equality only for X is an element of K(4). Kuhnel observed that this implies f(0)(f(0) - 11) >= -15 chi, with equality only for 2-neighborly members of K(4). Kuhnel also asked if there is a triangulated 4-manifold with f(0) = 15, chi = -4 (attaining equality in his lower bound). In this paper, guided by Kalai's theorem, we show that indeed there is such a triangulation. It triangulates the connected sum of three copies of the twisted sphere product S-3 (sic) S-1. Because of Kuhnel's inequality, the given triangulation of this manifold is a vertex-minimal triangulation. By a recent result of Effenberger, the triangulation constructed here is tight. Apart from the neighborly 2-manifolds and the infinite family of (2d + 3)-vertex sphere products Sd-1 X S-1 (twisted for d odd), only fourteen tight triangulated manifolds were known so far. The present construction yields a new member of this sporadic family. We also present a self-contained proof of Kalai's result. (C) 2011 Elsevier B.V. All rights reserved.

Item Type:	Journal Article
Publication:	Discrete Mathematics
Publisher:	Elsevier Science B.V.
Additional Information:	Copyright of this article belongs to Elsevier Science B.V.
Keywords:	Stacked spheres;Triangulated manifolds;Tight triangulations
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	30 May 2011 07:24
Last Modified:	30 May 2011 07:24
URI:	http://eprints.iisc.ac.in/id/eprint/38048

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