On stellated spheres and a tightness criterion for combinatorial manifolds

We introduce kk-stellated spheres and consider the class Wk(d)Wk(d) of triangulated dd-manifolds, all of whose vertex links are kk-stellated, and its subclass Wk∗(d), consisting of the (k+1)(k+1)-neighbourly members of Wk(d)Wk(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 Wk(d)Wk(d) for d≥2kd≥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 Wk∗(d) for d≥2k+2d≥2k+2. As another application, we prove that, when d≠2k+1d≠2k+1, all members of Wk∗(d) are tight. We also characterize the tight members of Wk∗(2k+1) in terms of their kkth 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 W1(d)W1(d) provide the equality case. This generalizes a result (the d=4d=4 case) due to Walkup and Kühnel. As a consequence, it is shown that every tight member of W1(d)W1(d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kühnel and Lutz asserting that tight homology manifolds should be strongly minimal.