Three‐Dimensional Pseudomanifolds  on Eight Vertices

Datta, Basudeb; Nilakantan, Nandini

doi:10.1155/2008/254637

Cited by 6 publications

(9 citation statements)

References 15 publications

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“…We start by recalling from [5] the necessary information on the topology of X i := N i , i = 1, 3, 4.…”

Section: Examplesmentioning

confidence: 99%

“…In this section we use methods and results developed in the paper so far to provide a complete characterization of the f -vectors of arbitrary simplicial triangulations of certain 3dimensional pseudomanifolds denoted by N 1 , N 3 , and N 4 in [4, Example 4]. We start by recalling from [4] the necessary information on the topology of X i := N i , i = 1, 3, 4.…”

Section: Examplesmentioning

confidence: 99%

“…Proof. To verify that (ii) implies (i), start with the triangulation N i of X i described in [4] (it satisfies g 1 (N i ) = 3 and g 2 (N i ) = 6) and follow the proof of Lemma 7.3 in Walkup's paper [27]. The triangulations N i (taken from [4]) and the "simple 3-trees" required to invoke this argument are listed at the end of this section.…”

Section: Examplesmentioning

confidence: 99%

See 2 more Smart Citations

Face numbers of pseudomanifolds with isolated singularities

Novik¹,

Swartz²

2012

MATH. SCAND.

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We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower and upper bound theorems when the singularities are also homologically isolated. We give formulas for the Hilbert function of a generic Artinian reduction of the face ring when the singularities are homologically isolated and for any pure two-dimensional complex. Some examples of spaces where the f -vector can be completely characterized are described. We also show that the Hilbert function of a generic Artinian reduction of the face ring of a simplicial complex with isolated singularities minus the h-vector of is a PL-topological invariant.

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“…We start by recalling from [5] the necessary information on the topology of X i := N i , i = 1, 3, 4.…”

Section: Examplesmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

See 1 more Smart Citation

Face numbers of pseudomanifolds with isolated singularities

Novik¹,

Swartz²

2012

MATH. SCAND.

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“…We do not know if there are any 3-dimensional complexes with relatively minimal g 2 , three or four singularities, but are not pseudomcompression bodies. Datta and Nilakantan determined all three-dimensional normal pseudomanifolds on eight vertices [9]. One of them, which they denoted N3, has five singularities, four projective planes and one torus.…”

Section: Few Singularitiesmentioning

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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“…Several classification and description of triangulated manifold and pseudomanifold has been done on the basis of f -vector and the value of g 2 . In [2] the d-pseudomanifold with d+4 vertices has been described completely followed by 9-vertex 3-pseudomanifold in [9]. In [15], it has been proved that only a finite number of combinatorial manifold is possible for a given upper bound of g 2 .…”

Section: Introductionmentioning

confidence: 99%

Characterization of $g_2$-minimal normal 3-pseudomanifolds with at most four singularities

Basak¹,

Gupta²,

Sarka³

2022

Preprint

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Let ∆ be a g 2 -minimal normal 3-pseudomanifold. A vertex in ∆ whose link is not a sphere is called a singular vertex. The combinatorial characterization of such ∆ is known when ∆ has at most two singular vertices [7]. In this article we have given the combinatorial characterization of such ∆, when ∆ contains either three singular vertices including one RP 2 singularity or four singular vertices including two RP 2 singularities. In both cases, we prove that ∆ is obtained from a one-vertex suspension of a surface and some boundary complex of 4-simplices by applying the combinatorial operations connected sum, vertex folding and edge folding.

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Three‐Dimensional Pseudomanifolds on Eight Vertices

Cited by 6 publications

References 15 publications

Face numbers of pseudomanifolds with isolated singularities

Face numbers of pseudomanifolds with isolated singularities

Three-dimensional normal pseudomanifolds with relatively few edges

Characterization of $g_2$-minimal normal 3-pseudomanifolds with at most four singularities

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