On k-stellated and k-stacked spheres

Abstract

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 ≥ 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 Σ k (d−1) (respectively, S k (d−1)). Thus,Let K 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 from K k (d) onto K k (d + 1) which is the inverse to the boundary map ∂ :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 W k (2k + 1) is F-tight if and only if it is k-stacked.Mathematics Subject Classification (2010): 52B05, 52B22, 52B11, 57Q15.In parallel with these classes of homology spheres, we also consider the classes Σ k (d) and S k (d) of k-shelled d-balls and k-stacked homology d-balls, respectively. We have the filtrationof the class of all shellable d-balls, and the comparable filtrationof the class of all homology d-balls. The standard d-ball B d d+1 is the unique (d + 1)-vertex triangulation of the d-dimensional ball. It may be described as the face complex of the d-dimensional geometric simplex. The standard ball B d d+1 is the unique member of Σ 0 (d) = S 0 (d). We also have the equality Σ 1 (d) = S 1 (d) and for all d ≥ k we have the inclusion Σ k (d) ⊆ S k (d). However, for each k ≥ 2, there are k-stacked balls which are not k-shelled.While a k-stellated d-sphere is defined as a triangulated d-sphere which may be obtained from S d d+2 by a finite sequence of bistellar moves of index < k, a k-shelled d-ball is a triangulated d-ball obtained from B d d+1 by a finite sequence of shelling moves of index < k. A k-stacked homology d-ball is a homology d-ball all whose faces of codimension k + 1 (i.e., dimension d − k − 1) are in its boundary. A k-stacked homology d-sphere is a homology d-sphere which may be represented as the boundary of a k-stacked (d + 1)-ball. The boundary of any k-shelled (d + 1)-ball is a k-stellated d-sphere. Conversely, when d ≥ 2k − 1, any k-stellated d-sphere may be represented as the boundary of a k-shelled (d + 1)-ball. A homology ball is k-shelled if and only if it is k-stacked and shellable. Each k-stacked homology (respectively k-shelled) ball is the antistar of a vertex in a k-stacked homology (respectively k-stellated) sphere. Murai and Nevo [17] proved that, when d ≥ 2k, for any k-stacked homology d-sphere S, there is a unique k-stacked homology (d + 1)-ball S whose boundary is S. The ball S has a natural and intrinsic description in terms of the combinatorics of S. We point out the Murai-Nevo theorem is an immediate consequence o...

Lower Bound Theorems and a Generalized Lower Bound Conjecture for Balanced Simplicial Complexes

“…We should stress that while the Generalized Lower Bound Conjecture for the class of simplicial polytopes is now a theorem (the inequalities of the GLBC form a part of the g-theorem, and the treatment of equality cases was recently completed in [32]), the balanced GLBC is at present a conjecture, even for the class of balanced polytopes. We verify the "easy" part of the equality case of this conjecture, see Theorem 5.10, and prove that, in analogy with the results from [6,33], manifolds with the balanced r-stacked property have a unique r-stacked cross-polytopal decomposition, see Theorem 5.17.…”

Tight triangulations of closed 3-manifolds

“…The notion of stackedness was introduced by Walkup [23] and McMullen & Walkup [18] in the context of triangulated spheres. The close relationship between stackedness and tightness has been highlighted in [1,5,6,19]. This is further borne out by Theorem 4.8 of this article.…”

“…Thus, all F-orientable, neighbourly, stacked, closed, triangulated 3-manifolds are F-tight (for any field F). Note that for d ≥ 4, the notions of stackedness and local stackedness coincide for triangulated manifolds ( [13,5,20]). In conjunction with Proposition 1.6, this shows that it is stackedness (as opposed to local stackedness) which seems to be central to tightness.…”

A Construction Principle for Tight and Minimal Triangulations of Manifolds

“…For this example, we have (m 0 , m 1 , m 2 ) = (41, 20, 1) (see appendix). Also, we have the following treetypes: (3,2), (3,4), (4, 2), (4, 3), (4, 4)}, ∆(σ 2 , τ 2 ) = {(1, 3), (2, 3), (2, 4), (4, 1), (4, 3), (4, 4)}, where D = {(σ 1 , τ 1 ), (σ 2 , τ 2 )} is the 2-deck of permutations of {0, 1, 2, 3}. It can be seen that we must have, σ 1 = (1, 2, 0, 3), τ 1 = (1, 0, 3, 2); σ 2 = (1, 0, 2, 3), τ 2 = (0, 2, 1, 3).…”

“…∆(σ 3 , τ 3 ) = {(2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (4, 5), (5, 2)}, ∆(σ 2 , τ 2 ) = {(1, 1), (1,2), (1,4), (3,2), (3,4), (4, 2), (5, 1), (5,2), (5,4), (5, 5)}, ∆(σ 1 , τ 1 ) = {(2, 4), (3,3), (3,4), (4, 3), (4, 4), (4, 5), (5, 1), (5,3), (5,4), (…”