2006/10/27 by Bhaskar Bagchi, Bagchi, Bhaskar, Basudeb Datta +1
Computer Science · Mathematics · #57Q15 #57R05 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric Topology (math.GT) #Topological and Geometric Data Analysis #math.CO #math.GT #msc:57Q15 #msc:57R05
paper · pdf · doi:10.48550/arxiv.math/0610829
15 pages, Revised, To appear in `Journal of Combinatorial Theory, Ser. A'
openalex publication_date 2006/10/27 · arxiv created 2007/10/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For integers d ≥ 2 and ε= 0 or 1, let S1, d - 1(ε) denote the sphere product S1 × Sd - 1 if ε= 0 and the twisted Sd - 1 bundle over S1 if ε= 1. The main results of this paper are: (a) if d ≡ ε (mod 2) then S1, d - 1(ε) has a unique minimal triangulation using 2d + 3 vertices, and (b) if d ≡ 1 - ε (mod 2) then S1, d - 1(ε) has minimal triangulations (not unique) using 2d + 4 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 S1, d - 1(ε) has at most one (2d + 3)-vertex triangulation (one if d ≡ ε (mod 2), 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 d ≥ 3, 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.