On stellated spheres, shellable balls, lower bounds and a combinatorial criterion for tightness

We introduce the k-stellated (combinatorial) spheres and compare and contrast them with kstacked (triangulated) spheres. It is shown that for d ≥ 2k, any k-stellated sphere of dimension d bounds a unique and canonically defined k-stacked ball. In parallel, any k-stacked polytopal sphere of dimension d ≥ 2k bounds a unique and canonically defined k-stacked (polytopal) ball, which answers a question of McMullen. We consider the class Wk(d) of combinatorial d-manifolds with k-stellated links. For d ≥ 2k + 2, any member of Wk(d) bounds a unique and canonically defined “k-stacked” (d+ 1)-manifold. We introduce the mu-vector of simplicial complexes, and show that the mu-vector of any 2neighbourly simplicial complex dominates its vector of Betti numbers componentwise, and the two vectors are equal precisely when the complex is tight. When d ≥ 2k, we are able to estimate/compute certain alternating sums of the mu-numbers of any 2-neighbourly member of Wk(d). This leads to a lower bound theorem for such triangulated manifolds. As an application, it is shown that any (k + 1)-neighbourly member of Wk(d) is tight, subject only to an extra condition on the k Betti number in case d = 2k + 1. This result more or less settles a recent conjecture of Effenberger, and it also provides a uniform and conceptual tightness proof for all the known tight triangulated manifolds, with only two exceptions. It is shown that any polytopal upper bound sphere of odd dimension 2k+ 1 belongs to the class Wk(2k+1), thus generalizing a theorem (the k = 1 case) due to Perles. This shows that the case d = 2k + 1 is indeed exceptional for the tightness theorem. 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. Mathematics Subject Classification (2010): 57Q15, 57R05, 52B05.