On stellated spheres and a tightness criterion for combinatorial manifolds

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 ≥ 2 k . 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 ≥ 2 k + 2 . As another application, we prove that, when d ≠ 2 k + 1 , all members of W k ∗ ( d ) are tight. We also characterize the tight members of W k ∗ ( 2 k + 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. o 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 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 homology manifolds should be strongly minimal.