Separation index of graphs and stacked 2-spheres

Burton, Benjamin A.; Datta, Basudeb; Singh, Nitin; Spreer, Jonathan

doi:10.1016/j.jcta.2015.07.001

Cited by 8 publications

(10 citation statements)

References 16 publications

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“…This bound was proved by Novic and Swartz in [15]. Burton et al proved in [6] that if the equality holds in this inequality then M is neighbourly and locally stacked. (Actually, these authors stated this result for F = Z 2 , but their argument goes through for all fields F.) In [1], the first author proved that the equality holds in this inequality if and only if M is neighbourly and stacked.…”

Section: Introductionmentioning

confidence: 88%

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A characterization of tightly triangulated 3-manifolds

Bagchi

Datta

Spreer

2017

European Journal of Combinatorics

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For a field F, the notion of F-tightness of simplicial complexes was introduced by Kühnel. Kühnel and Lutz conjectured that any F-tight triangulation of a closed manifold is the most economic of all possible triangulations of the manifold. The boundary of a triangle is the only F-tight triangulation of a closed 1-manifold. A triangulation of a closed 2-manifold is F-tight if and only if it is F-orientable and neighbourly. In this paper we prove that a triangulation of a closed 3-manifold is F-tight if and only if it is F-orientable, neighbourly and stacked. In consequence, the Kühnel-Lutz conjecture is valid in dimension ≤ 3.MSC 2010 : 57Q15, 57R05.

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Section: Introductionmentioning

confidence: 88%

“…Proof of Corollary 1.5. If (f 0 (M ) − 4)(f 0 (M ) − 5) = 20β 1 (M ; F) then Theorem 1.3 of [6] says that M must be neighbourly and locally stacked. Therefore, the 'if part' follows from Theorem 2.24 of [2].…”

Section: Proofsmentioning

confidence: 99%

A characterization of tightly triangulated 3-manifolds

Bagchi

Datta

Spreer

2017

European Journal of Combinatorics

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“…The τ -vector is called the normalized σ-vector by Murai and Novik in [43], where it is denotedσ. It is a variation of the σ-vector introduced by Bagchi and Datta [8] and studied in [7,16]. Its original motivation was the study of tight triangulations of manifolds, that is, triangulations with the property that all the homomorphisms induced in homology by inclusions of induced subcomplexes are injective (see Section 6.2, in particular Theorem 6.2 for more details): Moreover, the τ -vector is also interesting from a more combinatorial viewpoint.…”

Section: Previous Workmentioning

confidence: 99%

“…Remark 2.5. In [7], [8] and [16],β 0 is defined to be equal to β 0 − 1. This coincides with our convention, see Section 2.1, except for the empty complex where their convention givesβ 0 = −1 and ours givesβ 0 = 0.…”

Section: The τ -Vectormentioning

confidence: 99%

Average Betti Numbers of Induced Subcomplexes in Triangulations of Manifolds

Codenotti

Spreer²,

Santos³

2020

Electron. J. Combin.

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We study a variation of Bagchi and Datta's $\sigma$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of $C$. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sphere maximizes these invariants among triangulated spheres with a given $f$-vector. For the first entry of $\sigma$, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on $\sigma$ can be used to obtain lower bounds on the $f$-vector of triangulated $4$-manifolds with transitive symmetry on vertices and prescribed vector of Betti numbers.

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“…Case n = 9: We have 14 triangulations of the two-sphere with eight vertices with the following σ 0 -values. Hence, to satisfy Equation (6.1) only the seven triangulations with σ 0 -value equal to zero can be considered (these are precisely the seven stacked eight-vertex two-spheres, see [5] for a more general observation on the σ 0 -value of two-sphere triangulations). Amongst these seven triangulations only the triangulation presented in Figure 6.1 satisfies Property T 1 and thus any tight combinatorial three-manifold M with β 1 (M, F) = 1 must have nine isomorphic vertex links of that type.…”

Section: ( − 1)-connected (2 + 1)-manifoldsmentioning

confidence: 99%

A necessary condition for the tightness of odd-dimensional combinatorial manifolds

Spreer

2016

European Journal of Combinatorics

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We present a necessary condition for ( − 1)-connected combinatorial (2 + 1)-manifolds to be tight. As a corollary, we show that there is no tight combinatorial three-manifold with Betti number at most two other than the boundary of the four-simplex and the nine-vertex triangulation of the three-dimensional Klein bottle. MSC 2010: 57Q15; 57N10; 05A19where (a) n = a ⋅ (a + 1) ⋅ (a + 2) ⋅ . . . ⋅ (a + n − 1) denotes the Pochhammer symbol.As of today, the known cases of equality in (1.1) are the boundary of the simplex ( ≥ 1, β = 0) and the 13-vertex triangulation of SU (3) SO(3) ( = 2 and β = 1).As a direct consequence any (F-)tight connected combinatorial three-manifold M with β 1 (M, F) ≤ 2 cannot have more than 12 vertices. Together with further results presented in Section 6 and extended computer experiments this leads to the following.Corollary 1.2. The boundary of the simplex and the nine-vertex three-dimensional Klein Bottle S 2 S 1 are the only tight combinatorial three-manifolds with first Betti number at most two.denotes the number of i-dimensional faces of M . The zerodimensional faces of M are called vertices, the one-dimensional faces are called edges and the d-dimensional faces are referred to as facets. The set of vertices of M will be denoted by V (M ) or just V if M is given by the context. We call M k-neighbourly, if f k−1 = f0 k , i.e., if it contains all possible (k − 1)-dimensional faces. An n-vertex combinatorial d-manifold M distinct from the boundary of the (d + 1)-simplex can be at most (⌊ d+2 2 ⌋)-neighbourly. In this case the f -vector of an odd-dimensional combinatorial manifold M is already determined to be the one of the boundary complex of the (even-dimensional) cyclic (d + 1)-polytope with n vertices. This statement is known as the Upper Bound Theorem due to Novik [17] and Novik and Swartz [18]. Given a combinatorial manifold M with vertex set V (M ) and W ⊂ V (M ), the simplicial complex M [W ] = {σ ∈ M V (σ) ⊂ W }, i.e., the simplicial complex of all faces of M with vertex set in W , is called the sub-complex of M induced by W . TightnessTightness is a condition on subsets of Euclidean space generalising the notion of convexity: an object is tight if it is "as convex as possible", i. e., as simple as possible, given its topological constraints. More precisely we have the following. Definition 2.1 (Tightness [12]). A compact connected subset M ⊂ E d is called k-tight with respect to a field F if for every open or closed half space h ⊂ E d the induced homomorphism H k (h ∩ M, F) → H k (M, F) is injective. If M ⊂ E d is k-tight with respect to F for all k, 0 ≤ k ≤ d, it is called tight.

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Separation index of graphs and stacked 2-spheres

Cited by 8 publications

References 16 publications

A characterization of tightly triangulated 3-manifolds

A characterization of tightly triangulated 3-manifolds

Average Betti Numbers of Induced Subcomplexes in Triangulations of Manifolds

A necessary condition for the tightness of odd-dimensional combinatorial manifolds

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