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On stellated spheres and a tightness criterion for combinatorial manifolds

Bagchi, Bhaskar and Datta, Basudeb (2014) On stellated spheres and a tightness criterion for combinatorial manifolds. In: EUROPEAN JOURNAL OF COMBINATORICS, 36 . pp. 294-313.

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Official URL: http://dx.doi.org/10.1016/j.ejc.2013.07.018

Abstract

We introduce k-stellated spheres and consider the class W-k(d) of triangulated d-manifolds, all of whose vertex links are k-stellated, and its subclass W-k*; (d), consisting of the (k + 1)-neighbourly members of W-k(d). We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of W-k(d) for d >= 2k. As a consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, and we determine the integral homology type of members of W-k*(d) for d >= 2k + 2. As another application, we prove that, when d not equal 2k + 1, all members of W-k*(d) are tight. We also characterize the tight members of W-k*(2k + 1) in terms of their kth Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds. We also prove a lower bound theorem for homology manifolds in which the members of W-1(d) provide the equality case. This generalizes a result (the d = 4 case) due to Walkup and Kuhnel. As a consequence, it is shown that every tight member of W-1 (d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kuhnel and Lutz asserting that tight homology manifolds should be strongly minimal. (C) 2013 Elsevier Ltd. All rights reserved.

Item Type: Journal Article
Additional Information: Copyright for this article belongs to the ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD, ENGLAND
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Depositing User: Id for Latest eprints
Date Deposited: 11 Feb 2014 10:09
Last Modified: 11 Feb 2014 10:09
URI: http://eprints.iisc.ac.in/id/eprint/48355

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