Tight triangulations of closed 3-manifolds

Abstract

A triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the Kühnel-Lutz conjecture is valid in dimension three for fields of odd characteristic.Next let F be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is F-tight. For closed, triangulated 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of the existence of an F-tight, non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an F-tight triangulation of a closed 3-manifold has n vertices and first Betti number β 1 , then (n − 4)(617n − 3861) ≤ 15444β 1 . Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.MSC 2010 : 57Q15, 57R05.

A characterization of tightly triangulated 3-manifolds

“…The special case of Theorem 1.2, where char(F) = 2, was proved in our previous paper [4]. In this paper we conjectured [4, Conjecture 1.12] the validity of Theorem 1.2 in general.…”

“…Let M be an F-tight triangulated closed d-manifold, d ≤ 3. By Lemma 2.2 in [4], M is neighbourly. But the boundary complex of the triangle is the only neighbourly triangulated closed 1-manifold.…”

Separation index of graphs and stacked 2-spheres

“…In fact, such a triangulation, if it exists, must fulfil very strong conditions. For instance, following a recent result of Bagchi and the second and fourth authors [4], any tight triangulated 3-manifold with first Betti number smaller than 189 must be tight-neighbourly.…”

“…For instance, a triangulated 2-sphere is flag if and only if it has no induced 3-cycle (i.e., there is no set of three vertices spanning three edges but not a triangle). Since a flag 2-sphere is not a connected sum of two triangulated 2-spheres, a flag 2-sphere is also called primitive [4].…”

A Construction Principle for Tight and Minimal Triangulations of Manifolds

“…For d = 3, the neighbourly members of K(d) are tight. Theorem 2.3 is not true for d = 3 (see for example [6,Example 6.2]). However, the following more restrictive version holds in the three-dimensional case.…”