2012/08/07 by Bhaskar Bagchi, Bagchi, Bhaskar, Basudeb Datta +1
Computer Science · Mathematics · #52B05 #52B11 #52B22 #57Q15 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · doi:10.48550/arxiv.1208.1389
openalex publication_date 2012/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the class Σk(d) of k-stellated (combinatorial) spheres of dimension d (0 ≤ k ≤ d + 1) and compare and contrast it with the class \cal Sk(d) (0 ≤ k ≤ d) of k-stacked homology d-spheres. We have Σ1(d) = \cal S1(d), and Σk(d) ⊆ \cal Sk(d) for d ≥ 2k - 1. However, for each k ≥ 2 there are k-stacked spheres which are not k-stellated. The existence of k-stellated spheres which are not k-stacked remains an open question. We also consider the class \cal Wk(d) (and \cal Kk(d)) of simplicial complexes all whose vertex-links belong to Σk(d - 1) (respectively, \cal Sk(d - 1)). Thus, \cal Wk(d) ⊆ \cal Kk(d) for d ≥ 2k, while \cal W1(d) = \cal K1(d). Let \cal Kk(d) denote the class of d-dimensional complexes all whose vertex-links are k-stacked balls. We show that for d≥ 2k + 2, there is a natural bijection M ↦ M from \cal Kk(d) onto \cal Kk(d + 1) which is the inverse to the boundary map ∂ \colon \cal Kk(d + 1) → \cal Kk(d).