Tight triangulations of closed 3-manifolds

Bagchi, Bhaskar and Datta, Basudeb and Spreer, Jonathan (2016) Tight triangulations of closed 3-manifolds. In: EUROPEAN JOURNAL OF COMBINATORICS, 54 . pp. 103-120.

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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 Kuhnel-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 beta(1), then (n - 4) (617n - 3861) <= 15444 beta(1). Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra. (C) 2015 Elsevier Ltd. All rights reserved.

Item Type:	Journal Article
Publication:	EUROPEAN JOURNAL OF COMBINATORICS
Publisher:	ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
Additional Information:	Copy right for this article belongs to the ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD, 24-28 OVAL RD, LONDON NW1 7DX, ENGLAND
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	02 Apr 2016 07:03
Last Modified:	02 Apr 2016 07:03
URI:	http://eprints.iisc.ac.in/id/eprint/53573

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