Minimal triangulations of sphere bundles over the circle

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• 作者：Bhaskar Bagchi ; Basudeb Datta
• 关键词：Triangulated manifolds ; Stacked spheres ; Non-simply connected manifolds
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2008
• 出版时间：July 2008
• 年：2008
• 卷：115
• 期：5
• 页码：737-752
• 全文大小：213 K

文摘

For integers d2 and ε=0 or 1, let S^1,d−1(ε) denote the sphere product S¹×S^d−1 if ε=0 and the twisted sphere product if ε=1. The main results of this paper are: (a) if then S^1,d−1(ε) has a unique minimal triangulation using 2d+3 vertices, and (b) if then S^1,d−1(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,d−1(ε) has at most one (2d+3)-vertex triangulation (one if , zero otherwise), in sharp contrast, the number of non-isomorphic (2d+4)-vertex triangulations of these d-manifolds 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 d3, there is a unique (2d+3)-vertex simplicial complex which triangulates a non-simply 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 non-simply connected closed d-manifold 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.