Lower bound theorem for normal pseudomanifolds

Bagchi, Bhaskar; Datta, Basudeb

doi:10.1016/j.exmath.2008.06.002

Cited by 29 publications

(58 citation statements)

References 14 publications

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“…In the rest of the paper, we denote by G d the set of complexes that is either the join of ∂σ i and ∂σ d−i , where 2 ≤ i ≤ d − 2, or the join of ∂σ d−2 and a cycle. The following lemma is a special case of Theorem 1(a) in [4]. We give a proof for the sake of completeness.…”

Section: Retriangulations Of Simplicial Complexesmentioning

confidence: 90%

A characterization of homology manifolds with g2≤ 2

Zheng

2018

Journal of Combinatorial Theory, Series A

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We characterize homology manifolds with g 2 ≤ 2. Specifically, using retriangulations of simplicial complexes, we give a short proof of Nevo and Novinsky's result on the characterization of homology (d − 1)-spheres with g 2 = 1 for d ≥ 5 and extend it to the class of normal pseudomanifolds. We proceed to prove that every prime homology manifold with g 2 = 2 is obtained by centrally retriangulating a polytopal sphere with g 2 ≤ 1 along a certain subcomplex. This implies that all homology manifolds with g 2 = 2 are polytopal spheres. arXiv:1607.05757v2 [math.CO]

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Section: Retriangulations Of Simplicial Complexesmentioning

confidence: 90%

A characterization of homology manifolds with g2≤ 2

Zheng

2018

Journal of Combinatorial Theory, Series A

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“…(More generally, S d n usually denotes an n-vertex triangulation of the d-sphere.) From the definition of a stacked sphere, one can deduce the following (this also follows from [4,Lemmas 4…”

Section: Preliminaries On Stacked and Tight Triangulationsmentioning

confidence: 90%

Tight triangulations of closed 3-manifolds

Bagchi

Datta

Spreer

2016

European Journal of Combinatorics

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A triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the Kühnel-Lutz conjecture is valid in dimension three for fields of odd characteristic.Next let F be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is F-tight. For closed, triangulated 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of the existence of an F-tight, non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an F-tight triangulation of a closed 3-manifold has n vertices and first Betti number β 1 , then (n − 4)(617n − 3861) ≤ 15444β 1 . Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.MSC 2010 : 57Q15, 57R05.

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“…A bijection ψ : σ 1 → σ 2 is said to be admissible (cf. [3]) if for all vertices x ∈ σ 1 the edge distance between x and ψ(x) in the graph of ∆ is at least three. If ψ is an admissible bijection between σ 1 and σ 2 then we can form a new simplicial complex by identifying all faces ρ 1 ⊆ σ 1 , ρ 2 ⊆ σ 2 such that ψ maps ρ 1 bijectively onto ρ 2 .…”

Section: Constructionsmentioning

confidence: 99%

Three-dimensional normal pseudomanifolds with relatively few edges

Basak

Swartz

2020

Advances in Mathematics

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Let ∆ be a d-dimensional normal pseudomanifold, d ≥ 3. A relative lower bound for the number of edges in ∆ is that g 2 of ∆ is at least g 2 of the link of any vertex. When this inequality is sharp ∆ has relatively minimal g 2 . For example, whenever the one-skeleton of ∆ equals the one-skeleton of the star of a vertex, then ∆ has relatively minimal g 2 . Subdividing a facet in such an example also gives a complex with relatively minimal g 2 . We prove that in dimension three these are the only examples. As an application we determine the combinatorial and topological type of 3-dimensional ∆ with relatively minimal g 2 whenever ∆ has two or fewer singularities. The topological type of any such complex is a pseudocompression body, a pseudomanifold version of a compression body.Complete combinatorial descriptions of ∆ with g 2 (∆) ≤ 2 are due to Kalai [12] (g 2 = 0), Nevo and Novinsky [13] (g 2 = 1) and Zheng [21] (g 2 = 2). In all three cases ∆ is the boundary of a simplicial polytope. Zheng observed that for all d ≥ 0 there are triangulations of S d * RP 2 with g 2 = 3. She asked if this is the only nonspherical topology possible for g 2 (∆) = 3. As another application of relatively minimal g 2 we give an affirmative answer when ∆ is 3-dimensional.

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Lower bound theorem for normal pseudomanifolds

Cited by 29 publications

References 14 publications

A characterization of homology manifolds with g2≤ 2

A characterization of homology manifolds with g2≤ 2

Tight triangulations of closed 3-manifolds

Three-dimensional normal pseudomanifolds with relatively few edges

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