16 research outputs found
Combinatorial triangulations of homology spheres
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
For integers and or 1, let
denote the sphere product if and the
twisted bundle over if . The main results of
this paper are: (a) if (mod 2) then has a unique minimal triangulation using vertices, and
(b) if (mod 2) then has
minimal triangulations (not unique) using 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
has at most one -vertex triangulation (one if
(mod 2), zero otherwise), in sharp contrast, the number of non-isomorphic -vertex triangulations of these -manifolds grows exponentially with
for either choice of . The result in (a), as well as the minimality
part in (b), is a consequence of the following result: (c) for ,
there is a unique -vertex simplicial complex which triangulates a
non-simply connected closed manifold of dimension . 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 -manifold requires at least 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 -stellated and -stacked spheres
We introduce the class of -stellated (combinatorial) spheres
of dimension () and compare and contrast it with the
class () of -stacked homology -spheres.
We have , and for . However, for each there are
-stacked spheres which are not -stellated. The existence of -stellated
spheres which are not -stacked remains an open question.
We also consider the class (and ) of
simplicial complexes all whose vertex-links belong to
(respectively, ). Thus, for , while . Let
denote the class of -dimensional complexes all whose
vertex-links are -stacked balls. We show that for , there is a
natural bijection from onto which is the inverse to the boundary map .Comment: Revised Version. Theorem 2.24 is new. 18 pages. arXiv admin note:
substantial text overlap with arXiv:1102.085