On stellated spheres and a tightness criterion for combinatorial manifolds

We introduce the k-stellated spheres and consider the class Wk(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 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 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 Wk(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 6= 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 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 W1(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 W1(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. Mathematics Subject Classification (2010): 57Q15, 52B05.