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# On k-stellated and k-stacked spheres

Bagchi, Bhaskar and Datta, Basudeb (2013) On k-stellated and k-stacked spheres. In: Discrete Mathematics, 313 (20). pp. 2318-2329. PDF Disc_Math_313-20_2318_2013.pdf - Published Version Restricted to Registered users only Download (467kB) | Request a copy
Official URL: http://dx.doi.org/10.1016/j.disc.2013.06.010

## Abstract

We introduce the class Sigma(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 E-1(d) = S-1(d), and Sigma(k)(d) subset of S-k(d) ford >= 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 W-k(d) (and K-k(d)) of simplicial complexes all whose vertex-links belong to Sigma(k)(d - 1) (respectively, S-k(d - 1)). Thus, W-k(d) subset of K-k(d) for d >= 2k, while W-1(d) = K-1(d). Let (K) over bar (k)(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) over bar from K-k(d) onto (K) over bar (k)(d + 1) which is the inverse to the boundary map partial derivative: (K) over bar (k)(d + 1) -> (K) over bar (k)(d). Finally, 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(2k + 1) is F-tight if and only if it is k-stacked.

Item Type: Journal Article Copyright of this article belongs to Elsevier Science. Stacked Spheres; Homology Spheres; Shelling Moves; Bistellar Moves Division of Physical & Mathematical Sciences > Mathematics Francis Jayakanth 29 Oct 2013 05:40 29 Oct 2013 05:40 http://eprints.iisc.ac.in/id/eprint/47470 View Item