Paper Info
Title
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 ) ( 617 n - 3861 ) ¿ 15444 β 1 . Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.
Year
DOI
Venue
2016
10.1016/j.ejc.2015.12.006
European Journal of Combinatorics
Field
DocType
Volume
Discrete mathematics,Combinatorics,Betti number,Vertex (geometry),Triangulation,Triangulation (social science),Conjecture,Mathematics,Manifold,Point set triangulation,Upper bound theorem
Journal
54
Issue
ISSN
Citations 
C
European Journal of Combinatorics 54 (2016) 103--120
4
PageRank 
References 
Authors
0.60
7
3
Name
Order
Citations
PageRank
Bhaskar Bagchi17015.28
Basudeb Datta26413.91
Jonathan Spreer34711.46