Minimal triangulations of sphere bundles over the circle

Abstract

For integers d 2 and ε = 0 or 1, let S 1,d−1 (ε) denote the sphere product S 1 × S d−1 if ε = 0 and the twisted sphere product S 1 S d−1 if ε = 1. The main results of this paper are: (a) if d ≡ ε (mod 2) then S 1,d−1 (ε) has a unique minimal triangulation using 2d + 3 vertices, and (b) if d ≡ 1 − ε (mod 2) 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 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.

On Walkup’s class K(d) and a minimal triangulation of (S3Bagchi1, Datta2 2011Discrete MathematicsSelf Cite4000View full textAdd to dashboardCite

“…Indeed, there is a unique 11-vertex 4-manifold with χ = 0 (cf. [2]): it triangulates S 3 × S 1 . In [8], Kühnel asked if the next feasible case n = 15, χ = −4 can be realized.…”

“…It is also easy to see that Y ∈ K(d) if and only if  Y ∈ K(d) (cf. Lemma 2.6 in [2]). We use these results in the following proof.…”

“…But the combinatorial connected sum of stacked spheres is easily seen to be a stacked sphere (cf. Lemma 2.5 in [2]). So, X is a stacked sphere.…”

“…In Lemma 1.3 of [2], we have shown (in particular) that if Y is a connected triangulated closed manifold of dimension d ≥ 3, with a vertex set S on which Y induces an S d−1 d+1 , the above construction can be reversed. Namely, then there exists a unique triangulated closed d-manifold  Y , together with an admissible map ψ: σ 1 → σ 2 , such that Y = (  Y ) ψ , and S = σ 1 ≈ σ 2 .…”

“…This orientation gives a coherent orientation on S 4 30 in which b ′ triangulated manifolds (i.e., the so-called regular cases in Heawood's map color theorem; cf. [13]) and (c) the (2d + 3)-vertex combinatorial d-manifold K d 2d+3 due to Kühnel [2,8]: it triangulates S d−1 S 1 if d is odd and S d−1 × S 1 if d is even.…”