# On stellated spheres and a tightness criterion for combinatorial manifolds

We introduce the $k$-stellated spheres and consider the class ${\cal W} kd$ of triangulated $d$-manifolds all whose vertex links are $k$-stellated, and its subclass ${\cal W}^{\ast} kd$ consisting of the $k+1$-neighbourly members of ${\cal W} kd$. 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 ${\cal W} kd$ for $d\geq 2k$. 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 ${\cal W}^{\ast} kd$ for $d \geq 2k+2$. As another application, we prove that, when $d eq 2k+1$, all members of ${\cal W}^{\ast} kd$ are tight. We also characterize the tight members of ${\cal W}^{\ast} k2k + 1$ in terms of their $k^{ m th}$ 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 ${\cal W} 1d$ provide the equality case. This generalises a result the $d=4$ case due to Walkup and Kuehnel. As a consequence, it is shown that every tight member of ${\cal W} 1d$ is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kuehnel and Lutz asserting that tight triangulated manifolds should be strongly minimal.