On -stellated and -stacked spheres

We introduce the class Σ k ( d ) of k -stellated (combinatorial) spheres of dimension d ( 0 ≤ k ≤ d + 1 ) and compare and contrast it with the class S k ( d ) ( 0 ≤ k ≤ d ) of k -stacked homology d -spheres. We have Σ 1 ( d ) = S 1 ( d ) , and Σ k ( d ) ⊆ S k ( d ) for d ≥ 2 k − 1 . However, for each k ≥ 2 there are k -stacked spheres which are not k -stellated. For d ≤ 2 k − 2 , the existence of k -stellated spheres which are not k -stacked remains an open question. o consider the class W k ( d ) (and K k ( d ) ) of simplicial complexes all whose vertex-links belong to Σ k ( d − 1 ) (respectively, S k ( d − 1 ) ). Thus, W k ( d ) ⊆ K k ( d ) for d ≥ 2 k , while W 1 ( d ) = K 1 ( d ) . Let K ¯ k ( d ) denote the class of d -dimensional complexes all whose vertex-links are k -stacked balls. We show that for d ≥ 2 k + 2 , there is a natural bijection M ↦ M ¯ from K k ( d ) onto K ¯ k ( d + 1 ) which is the inverse to the boundary map ∂ : K ¯ k ( d + 1 ) → K k ( d ) . y, we complement the tightness results of our recent paper, Bagchi and Datta (2013)  [5], by showing that, for any field F , an F -orientable ( k + 1 ) -neighbourly member of W k ( 2 k + 1 ) is F -tight if and only if it is k -stacked.