On k-stellated and k-stacked spheres

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 Sk(d) (0 ≤ k ≤ d) of k-stacked homology d-spheres. We have Σ1(d) = S1(d), and Σk(d) ⊆ Sk(d) for d ≥ 2k− 1. However, for each k ≥ 2 there are k-stacked spheres which are not k-stellated. For d ≤ 2k− 2, the existence of k-stellated spheres which are not k-stacked remains an open question. We also consider the class Wk(d) (and Kk(d)) of simplicial complexes all whose vertex-links belong to Σk(d−1) (respectively, Sk(d−1)). Thus, Wk(d) ⊆ Kk(d) for d ≥ 2k, while W1(d) = K1(d). Let 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 7→ M from Kk(d) onto Kk(d+1) which is the inverse to the boundary map ∂ : Kk(d+ 1) → Kk(d). Finally, we complement the tightness results of our recent paper [5] by showing that, for any field F, an F-orientable (k+1)-neighborly member of Wk(2k+1) is F-tight if and only if it is k-stacked. Mathematics Subject Classification (2010): 52B05, 52B22, 52B11, 57Q15.