No abstract

No abstract

We introduce the k-stellated spheres and consider the class W k (d) of triangulated d-manifolds all whose vertex links are k-stellated, and its subclass W * k (d) consisting of the (k + 1)-neighbourly members of W k (d). We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of its Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of W k (d) for d ≥ 2k. As one consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, as well as determine the integral homology type of members of W * k (d) for d ≥ 2k + 2. As another application, we prove that, when d = 2k + 1, all members of W * k (d) are tight. We also characterize the tight members of W * k (2k + 1) in terms of their k th Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds.We also prove a lower bound theorem for triangulated manifolds in which the members of W 1 (d) provide the equality case. This generalises a result (the d = 4 case) due to Walkup and Kühnel. As a consequence, it is shown that every tight member of W 1 (d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kühnel and Lutz asserting that tight triangulated manifolds should be strongly minimal. (2010): 57Q15, 52B05. Mathematics Subject ClassificationDefinition 1.1. For 0 ≤ k ≤ d + 1, a d-dimensional simplicial complex X is said to be k-stellated if X may be obtained from S d d+2 by a finite sequence of bistellar moves, each of index < k. By convention, S d d+2 is the only 0-stellated simplicial complex of dimension d. Clearly, for 0 ≤ k ≤ l ≤ d + 1, k-stellated implies l-stellated. All k-stellated simplicial complexes are combinatorial spheres. By Pachner's theorem [25], the (d + 1)-stellated dspheres are precisely the combinatorial d-spheres.We also recall : Definition 1.2. For 0 ≤ k ≤ d + 1, a triangulated (d + 1)-dimensional ball B is said to be k-stacked if all the faces of B of codimension (at least) k + 1 lie in its boundary; i.e., if

No 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.