On stellated spheres and a tightness criterion for combinatorial manifolds

Abstract

We introduce the k-stellated spheres and consider the class W k (d) of triangulated d-manifolds all 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 its 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 one consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, as well as determine the integral homology type of members of W * k (d) for d ≥ 2k + 2. As another application, we prove that, when d = 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 k th 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 triangulated manifolds in which the members of W 1 (d) provide the equality case. This generalises a result (the d = 4 case) due to Walkup and Kühnel. 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 Kühnel and Lutz asserting that tight triangulated manifolds should be strongly minimal. (2010): 57Q15, 52B05. Mathematics Subject ClassificationDefinition 1.1. For 0 ≤ k ≤ d + 1, a d-dimensional simplicial complex X is said to be k-stellated if X may be obtained from S d d+2 by a finite sequence of bistellar moves, each of index < k. By convention, S d d+2 is the only 0-stellated simplicial complex of dimension d. Clearly, for 0 ≤ k ≤ l ≤ d + 1, k-stellated implies l-stellated. All k-stellated simplicial complexes are combinatorial spheres. By Pachner's theorem [25], the (d + 1)-stellated dspheres are precisely the combinatorial d-spheres.We also recall : Definition 1.2. For 0 ≤ k ≤ d + 1, a triangulated (d + 1)-dimensional ball B is said to be k-stacked if all the faces of B of codimension (at least) k + 1 lie in its boundary; i.e., if

A characterization of tightly triangulated 3-manifolds

“…If d = 3 then, by Theorem 1.2, M is stacked and hence is locally stacked. But any locally stacked, F-tight triangulated closed manifold is strongly minimal by Corollary 3.13 in [3]. So, we are done when d = 3.…”

Face numbers and the fundamental group

“…The µ ς -numbers were essentially introduced by Brehm and Kühnel [6] while the µ-numbers are due to Bagchi and Datta [3]. (Note that our µ 0 is slightly different than µ 0 of [3].) As can be seen from the above definition, these numbers are related to a version of the Morse theory for simplicial complexes (cf.…”

“…Moreover, if d ≥ 4, then g 2 (∆) = d+2 2 m(∆) if and only if ∆ is a stacked manifold. Our proof of Theorem 1.1 is based on studying the µ-numbers introduced by Bagchi and Datta [3], and, specifically, on the following result verified in [13] (see the proof of Theorem 5.3 there). We postpone the definition of the µ-numbers as well as several other definitions until the next section, and for now merely mention that the µ-numbers satisfy the Morse inequalities; in particular,…”

A generalized lower bound theorem for balanced manifolds

“…Bagchi and Datta [BD14] introduced the notion of µ-numbers. These numbers satisfy the following Morse-type inequalities: for any simplicial complex ∆, j k=0 (−1) j−k µ k (∆) ≥ j k=0 (−1) j−kβ k (∆) + (−1) j for all j ≥ 0.…”