16 research outputs found

    Combinatorial triangulations of homology spheres

    Get PDF
    Let Mu be an n-vertex combinatorial triangulation of a Ζ2-homology d-sphere. In this paper we prove that if n ≤ d+8 then Mu must be a combinatorial sphere. Further, if n=d+9 and M is not a combinatorial sphere then Mu cannot admit any proper bistellar move. Existence of a 12-vertex triangulation of the lens space L(3,1) shows that the first result is sharp in dimension three. In the course of the proof we also show that anyΖ2-acyclic simplicial complex on ≤7 vertices is necessarily collapsible. This result is best possible since there exist 8-vertex triangulations of the Dunce Hat which are not collapsible

    Minimal triangulations of sphere bundles over the circle

    Full text link
    For integers d2d \geq 2 and ϵ=0\epsilon = 0 or 1, let S1,d1(ϵ)S^{1, d - 1}(\epsilon) denote the sphere product S1×Sd1S^{1} \times S^{d - 1} if ϵ=0\epsilon = 0 and the twisted Sd1S^{d - 1} bundle over S1S^{1} if ϵ=1\epsilon = 1. The main results of this paper are: (a) if dϵd \equiv \epsilon (mod 2) then S1,d1(ϵ)S^{1, d - 1}(\epsilon) has a unique minimal triangulation using 2d+32d + 3 vertices, and (b) if d1ϵd \equiv 1 - \epsilon (mod 2) then S1,d1(ϵ)S^{1, d - 1}(\epsilon) has minimal triangulations (not unique) using 2d+42d + 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,d1(ϵ)S^{1, d - 1}(\epsilon) has at most one (2d+3)(2d + 3)-vertex triangulation (one if dϵd \equiv \epsilon (mod 2), zero otherwise), in sharp contrast, the number of non-isomorphic (2d+4)(2d + 4)-vertex triangulations of these dd-manifolds grows exponentially with dd for either choice of ϵ\epsilon. The result in (a), as well as the minimality part in (b), is a consequence of the following result: (c) for d3d \geq 3, there is a unique (2d+3)(2d + 3)-vertex simplicial complex which triangulates a non-simply connected closed manifold of dimension dd. This amazing simplicial complex was first constructed by K\"{u}hnel in 1986. Generalizing a 1987 result of Brehm and K\"{u}hnel, we prove that (d) any triangulation of a non-simply connected closed dd-manifold requires at least 2d+32d + 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.Comment: 15 pages, Revised, To appear in `Journal of Combinatorial Theory, Ser. A

    On kk-stellated and kk-stacked spheres

    Full text link
    We introduce the class Σk(d)\Sigma_k(d) of kk-stellated (combinatorial) spheres of dimension dd (0kd+10 \leq k \leq d + 1) and compare and contrast it with the class Sk(d){\cal S}_k(d) (0kd0 \leq k \leq d) of kk-stacked homology dd-spheres. We have Σ1(d)=S1(d)\Sigma_1(d) = {\cal S}_1(d), and Σk(d)Sk(d)\Sigma_k(d) \subseteq {\cal S}_k(d) for d2k1d \geq 2k - 1. However, for each k2k \geq 2 there are kk-stacked spheres which are not kk-stellated. The existence of kk-stellated spheres which are not kk-stacked remains an open question. We also consider the class Wk(d){\cal W}_k(d) (and Kk(d){\cal K}_k(d)) of simplicial complexes all whose vertex-links belong to Σk(d1)\Sigma_k(d - 1) (respectively, Sk(d1){\cal S}_k(d - 1)). Thus, Wk(d)Kk(d){\cal W}_k(d) \subseteq {\cal K}_k(d) for d2kd \geq 2k, while W1(d)=K1(d){\cal W}_1(d) = {\cal K}_1(d). Let Kˉk(d)\bar{{\cal K}}_k(d) denote the class of dd-dimensional complexes all whose vertex-links are kk-stacked balls. We show that for d2k+2d\geq 2k + 2, there is a natural bijection MMˉM \mapsto \bar{M} from Kk(d){\cal K}_k(d) onto Kˉk(d+1)\bar{{\cal K}}_k(d + 1) which is the inverse to the boundary map  ⁣:Kˉk(d+1)Kk(d)\partial \colon \bar{{\cal K}}_k(d + 1) \to {\cal K}_k(d).Comment: Revised Version. Theorem 2.24 is new. 18 pages. arXiv admin note: substantial text overlap with arXiv:1102.085
    corecore