2011/02/04 by Bhaskar Bagchi, Bagchi, Bhaskar, Basudeb Datta +1 · 1 citation
Computer Science · Mathematics · #52B05 #57Q15 #57R05 #Advanced Combinatorial Mathematics #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric Topology (math.GT) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1102.0856
openalex publication_date 2011/02/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We introduce the k-stellated spheres and compare and contrast them with k-stacked 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 ball. We consider the class \cal Wk(d) of combinatorial d-manifolds with k-stellated links. For d≥ 2k+2, any member of \cal 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 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≥ 2k, we are able to estimate/compute certain alternating sums of the mu-numbers of any 2-neighbourly member of \cal 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 \cal Wk(d) is tight, subject only to an extra condition on the kth 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 Wk(2k+1), thus generalizing a theorem due to Perles. This shows that the case d=2k+1 is indeed exceptional for the tightness theorem.