Bagchi, Bhaskar ; Datta, Basudeb (2008) Minimal triangulations of sphere bundles over the circle Journal of Combinatorial Theory, Series A, 115 (5). pp. 737752. ISSN 00973165

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S00973...
Related URL: http://dx.doi.org/10.1016/j.jcta.2007.09.005
Abstract
For integers d ≥ 2 and ε=0 or 1, let S^{1,d1}(ε) denote the sphere product S^{1}×S^{d1} if ε=0 and the twisted sphere product S^{1}X S^{d1} if ε=1. The main results of this paper are: (a) ifd ≡ ε (mod2) then S^{1,d1}(ε) has a unique minimal triangulation using 2d+3 vertices, and (b) if d ≡ε (mod2) then S^{1,d1}(ε) has minimal triangulations (not unique) using 2d+4 vertices. In this context, a minimal triangulation of a manifold is a triangulation using the least possible number of vertices. The second result confirms a recent conjecture of Lutz. The first result provides the first known infinite family of closed manifolds (other than spheres) for which the minimal triangulation is unique. Actually, we show that while S^{1,d1}(ε) has at most one (2d+3)vertex triangulation (one if d ≡ ε(mod 2), zero otherwise), in sharp contrast, the number of nonisomorphic (2d+4)vertex triangulations of these dmanifolds grows exponentially with d for either choice of ε. The result in (a), as well as the minimality part in (b), is a consequence of the following result: (c) for d ≥ 3, there is a unique (2d+3)vertex simplicial complex which triangulates a nonsimply connected closed manifold of dimension d. This amazing simplicial complex was first constructed by Kühnel in 1986. Generalizing a 1987 result of Brehm and Kühnel, we prove that (d) any triangulation of a nonsimply connected closed dmanifold requires at least 2d+3 vertices. The result (c) completely describes the case of equality in (d). The proofs rest on the Lower Bound Theorem for normal pseudomanifolds and on a combinatorial version of Alexander duality.
Item Type:  Article 

Source:  Copyright of this article belongs to Elsevier Science. 
Keywords:  Triangulated Manifolds; Stacked Spheres; NonSimply Connected Manifolds 
ID Code:  1005 
Deposited On:  25 Sep 2010 11:23 
Last Modified:  16 May 2016 12:11 
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