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

We introduce the $k$-stellated spheres and compare and contrast them with $k$-stacked spheres. It is shown that for $d \geq 2k$, 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 $d\geq 2k$ bounds a unique and canonically defined $k$-stacked ball. We consider the class ${\cal W}_k(d)$ of combinatorial $d$-manifolds with $k$-stellated links. For $d\geq 2k+2$, any member of ${\cal W}_k(d)$ 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 2-neighbourly simplicial complex dominates its vector of Betti numbers componentwise, and the two vectors are equal precisely when the complex is tight. When $d\geq 2k$, we are able to estimate/compute certain alternating sums of the mu-numbers of any 2-neighbourly member of ${\cal W}_k(d)$. This leads to a lower bound theorem for such triangulated manifolds. As an application, it is shown that any $(k+1)$-neighbourly member of ${\cal W}_k(d)$ is tight, subject only to an extra condition on the $k^{th}$ 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 ${\cal W}_k(2k+1)$, thus generalizing a theorem due to Perles. This shows that the case $d=2k+1$ is indeed exceptional for the tightness theorem.

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