A tightness criterion for homology manifolds with or without boundary

Abstract

A simplicial complex X is said to be tight with respect to a field F if X is connected and, for every induced subcomplex Y of X, the linear map H * (Y ; F) → H * (X; F) (induced by the inclusion map) is injective. This notion was introduced by Kühnel in [10]. In this paper we prove the following two combinatorial criteria for tightness. (a) Any (k + 1)neighbourly k-stacked F-homology manifold with boundary is F-tight. Also, (b) any F-orientable (k + 1)-neighbourly k-stacked F-homology manifold without boundary is F-tight, at least if its dimension is not equal to 2k + 1.The result (a) appears to be the first criterion to be found for tightness of (homology) manifolds with boundary. Since every (k + 1)neighbourly k-stacked manifold without boundary is, by definition, the boundary of a (k + 1)-neighbourly k-stacked manifold with boundary -and since we now know several examples (including two infinite families) of triangulations from the former class -theorem (a) provides us with many examples of tight triangulated manifolds with boundary.The second result (b) generalizes a similar result from [2] which was proved for a class of combinatorial manifolds without boundary. We believe that theorem (b) is valid for dimension 2k + 1 as well. Except for this lacuna, this result answers a recent question of Effenberger [8] affirmatively.

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.…”

Average Betti Numbers of Induced Subcomplexes in Triangulations of Manifolds

“…More precisely, for a k-neighborly complex all τ i with i k − 2 are zero (see Theorem 4.9) and for a k-stacked d-sphere all τ i with k i d − k − 1 are zero (see Theorem 5.6). The first result is due to Bagchi and Datta [8,Lemma 3.9], the second is due to Bagchi [6,Lemma 3]. Going further, we also look at manifolds that are almost 2-neighborly or almost 1-stacked and prove bounds for the entries of their τ -vectors (Theorems 3.12, 5.10 and 5.11).…”

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), (…”

“…. , m k ), DT ) is tight-neighbourly and triangulates (S d−1 × S 1 ) #β 1 or (S d−1 × − S 1 ) #β 1 , where β 1 is the first Betti number as in(2).…”