On Stacked Triangulated Manifolds

Datta, Basudeb; Murai, Shinji

doi:10.37236/6181

Cited by 11 publications

(16 citation statements)

References 15 publications

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“…The first main result of this paper (in Section 4) is an extension of Proposition 1.6: If a closed, triangulated 3-manifold is tight with respect to a field of odd characteristic then it must be (orientable, neighbourly and) stacked. This result answers Question 4.5 of [9] affirmatively, in the case of odd characteristic. As a consequence of Proposition 1.7 it follows that the Kühnel-Lutz conjecture is true in a special case, namely, if char(F) = 2 then any F-tight, closed, triangulated 3-manifold is strongly minimal.…”

Section: Introductionsupporting

confidence: 73%

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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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Section: Introductionsupporting

confidence: 73%

“…Example 6.2 shows that Proposition 2.10 does not hold in dimension 3. From [9] we know the following.…”

Section: Preliminaries On Stacked and Tight Triangulationsmentioning

confidence: 99%

Tight triangulations of closed 3-manifolds

Bagchi

Datta

Spreer

2016

European Journal of Combinatorics

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“…Each such manifold is obtained by starting with several disjoint boundary complexes of the d-simplex and repeatedly forming connected sums and/or handle additions. Similarly, each 1-stacked manifold with boundary (of dimension ≥ 2) is obtained by starting with a number of d-simplices and repeatedly forming connected unions and/or handle additions; see [18] for more details. Therefore, all 1-stacked manifolds without boundary are spheres or connected sums of sphere bundles over S 1 .…”

Section: R-stacked Manifolds and The Cases Of Equalitymentioning

confidence: 99%

Face enumeration on simplicial complexes

Klee

Novik

2016

The IMA Volumes in Mathematics and Its Applications

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“…By Theorem 1.2, X is stacked. But, by Corollary 3.13 (case d = 3) of [8], any stacked triangulation of a closed 3-manifold can be obtained from a stacked 3-sphere by a finite sequence of elementary handle additions. It is easy to see by an induction on the number k of handles added that X triangulates either S 3 (k = 0) or (S 2 × S 1 ) #k or (S 2 × − S 1 ) #k (k ≥ 1).…”

Section: Proofsmentioning

confidence: 99%

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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On Stacked Triangulated Manifolds

Cited by 11 publications

References 15 publications

Tight triangulations of closed 3-manifolds

Tight triangulations of closed 3-manifolds

Face enumeration on simplicial complexes

A characterization of tightly triangulated 3-manifolds

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