Tight triangulations of closed 3-manifolds

Bagchi, Bhaskar, Datta, Basudeb and Spreer, Jonathan (2016) Tight triangulations of closed 3-manifolds. European Journal of Combinatorics, 54 103-120. doi:10.1016/j.ejc.2015.12.006


Author Bagchi, Bhaskar
Datta, Basudeb
Spreer, Jonathan
Title Tight triangulations of closed 3-manifolds
Journal name European Journal of Combinatorics   Check publisher's open access policy
ISSN 0195-6698
1095-9971
Publication date 2016-05
Year available 2015
Sub-type Article (original research)
DOI 10.1016/j.ejc.2015.12.006
Open Access Status Not Open Access
Volume 54
Start page 103
End page 120
Total pages 18
Place of publication London, United Kingdom
Publisher Academic Press
Collection year 2016
Language eng
Formatted 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 β1, then (n−4)(617n−3861)≤15444β1. Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status UQ
Additional Notes Published online 31 December 2015

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
Official 2016 Collection
 
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