A characterization of tightly triangulated 3-manifolds

Bagchi, Bhaskar; Datta, Basudeb; Spreer, Jonathan

doi:10.1016/j.ejc.2016.10.005

Cited by 6 publications

(12 citation statements)

References 15 publications

(22 reference statements)

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“…This result does not hold for 3-manifolds since there are triangulations of 3-manifolds which are locally stacked but cannot be obtained by these operations (see e.g. [BDS,Example 6.2]). On the other hand, since the stackedness and the locally stackedness are equivalent in dimension ≥ 4 [BD2,MN], Kalai's result also characterizes stacked triangulations of connected closed manifolds of dimension ≥ 4.…”

Section: Introductionmentioning

confidence: 99%

On Stacked Triangulated Manifolds

Datta

Murai

2017

Electron. J. Combin.

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We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension d ≥ 4, if ∆ is a tight connected closed homology d-manifold whose ith homology vanishes for 1 < i < d − 1, then ∆ is a stacked triangulation of a manifold. These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.MSC 2010 : 57Q15, 57R20, 05C40.

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Section: Introductionmentioning

confidence: 99%

On Stacked Triangulated Manifolds

Datta

Murai

2017

Electron. J. Combin.

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“…In the combinatorial setting, tight triangulations of manifolds in dimensions ≤ 3 are known to be strongly minimal, i.e., they contain the minimum number of simplices in each dimension. This result is due to recent work by Bagchi, the second, and the fourth author [7], but had been conjectured a long time before in arbitrary dimensions by Kühnel and Lutz [20].…”

Section: Introductionmentioning

confidence: 63%

“…If X is a neighbourly member of K(3), then X is tight if and only if X is tight-neighbourly. Theorem 2.6 (Bagchi-Datta-Spreer [7]). A triangulated closed 3-manifold M is F-tight if and only if M is F-orientable, neighbourly and stacked.…”

Section: Tight and Tight-neighbourly Triangulationsmentioning

confidence: 99%

“…The treetype tuple (see (7)) that we lose by decreasing a i by one is (i + 1, j + 1) where a i + b j = d + 1. Similarly the treetype tuple we gain by increasing a d+1−i by one is (d + 2 − i, s + 1) where a d+1−i + b s = d (it becomes d + 1 after the increment).…”

Section: D−1 · ⌊D/2⌋! · ⌊(D − 1)/2⌋! Non-isomorphic Tight D-manifoldsmentioning

confidence: 99%

“…• In dimension three, it is known that, apart from the 3-sphere, at most boundaries of handlebodies can be tight. Moreover, an n-vertex triangulation of a 3-manifold M is tight if and only if the first Betti number of M satisfies β 1 (M, Z 2 ) = 1 20 (n−4)(n−5) [7]. In particular, in dimension three tightness can be decided -and tight triangulations of handlebodies can be recognised -in polynomial time [8].…”

Section: Introductionmentioning

confidence: 99%

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A Construction Principle for Tight and Minimal Triangulations of Manifolds

Burton

Datta

Singh

et al. 2016

Experimental Mathematics

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Tight triangulations are exotic, but highly regular objects in combinatorial topology. A triangulation is tight if all its piecewise linear embeddings into a Euclidean space are as convex as allowed by the topology of the underlying manifold. Tight triangulations are conjectured to be strongly minimal, and proven to be so for dimensions ≤ 3. However, in spite of substantial theoretical results about such triangulations, there are precious few examples. In fact, apart from dimension two, we do not know if there are infinitely many of them in any given dimension.In this paper, we present a computer-friendly combinatorial scheme to obtain tight triangulations, and present new examples in dimensions three, four and five. Furthermore, we describe a family of tight triangulated d-manifolds, with 2 d−1 ⌊d/2⌋!⌊(d−1)/2⌋! isomorphically distinct members for each dimension d ≥ 2. While we still do not know if there are infinitely many tight triangulations in a fixed dimension d > 2, this result shows that there are abundantly many.MSC 2010 : 57Q15, 57R05.

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Tight Complexes Are Golod

Iriye,

Kishimoto

2023

International Mathematics Research Notices

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Golodness of a simplicial complex is defined algebraically in terms of the Stanley–Reisner ring, and it has been a long-standing problem to find its combinatorial characterization. Tightness of a simplicial complex is a combinatorial analogue of a tight embedding of a manifold into Euclidean space, and has been studied in connection to minimal manifold triangulations. In this paper, by using an idea from toric topology, we prove that tight complexes are Golod, and as a corollary, we obtain that a triangulation of a closed connected orientable manifold is Golod if and only if it is tight, which is a combinatorial characterization of Golodness for manifold triangulations.

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A characterization of tightly triangulated 3-manifolds

Cited by 6 publications

References 15 publications

On Stacked Triangulated Manifolds

On Stacked Triangulated Manifolds

A Construction Principle for Tight and Minimal Triangulations of Manifolds

Tight Complexes Are Golod

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