On the number of simplices required to triangulate a Lie group

Duan, Haibao; Marzantowicz, Wacław; Zhao, Xuezhi

doi:10.1016/j.topol.2020.107559

Cited by 4 publications

(4 citation statements)

References 17 publications

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“…An important family of examples whose covering type can be estimated with our methods are non-simply connected Lie groups. Our results extend the results of [8] where estimates of covering type of all simple simply-connected compact Lie groups were obtained based on rational cohomology rings and the methods of [14].…”

Section: Covering Type Of Spaces With Finite Cyclic Fundamental Groupsupporting

confidence: 84%

“…Remark 4.15. While the specific numerical formulas may not be particularly nice or elucidating, we may still observe that triangulations of the quotient of the special unitary group by its centre require at least around 8 3 n 3 vertices. For comparison, a triangulation of SU (n) as estimated in [14, Corollary 3.7] requires at least around 2 3 n 3 vertices.…”

Section: Covering Type Of Spaces With Finite Cyclic Fundamental Groupmentioning

confidence: 94%

“…Final remarks. We must emphasize that similarly as it was done in [15] and [8] we can use the Lower Bound Theorem or its improved version the Generalized Lower Bound Theorem of [1] to estimate the number of simplices of given dimension i of a triangulation of manifold M (see [20] for a review of results). Indeed, this theorem estimates from below the coordinates f i , i ≥ 1, of the vector f (m) = (f 0 , f 1 , .…”

Section: 4mentioning

confidence: 99%

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Estimates of covering type and minimal triangulations based on category weight

Govc¹,

Marzantowicz²,

Pavešić³

2021

Preprint

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In a recent publication [14] we have introduced a new method, based on the Lusternik-Schnirelmann category and the cohomology ring of a space X, that yields lower bounds for the size of a triangulation of X. In this paper we present an important extension that takes into account the fundamental group of X. In fact, if π1(X) contains elements of finite order, then one can often find cohomology classes of high 'category weight', which in turn allow for much stronger estimates of the size of triangulations of X. We develop several weighted estimates and then apply our method to compute explicit lower bounds for the size of triangulations of orbit spaces of cyclic group actions on a variety of spaces including products of spheres, Stiefel manifolds, Lie groups and highly-connected manifolds.

show abstract

Section: Covering Type Of Spaces With Finite Cyclic Fundamental Groupsupporting

confidence: 84%

Section: Covering Type Of Spaces With Finite Cyclic Fundamental Groupmentioning

confidence: 94%

Section: 4mentioning

confidence: 99%

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Estimates of covering type and minimal triangulations based on category weight

Govc¹,

Marzantowicz²,

Pavešić³

2021

Preprint

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“…The question of the number of simplices required to triangulate a given manifold is often attacked by sophisticated cohomological methods involving characteristic classes (such arguments also often yield estimates on the minimal embedding dimension for the manifold). Surprisingly, much of this work is very recent [6], [7]. A main result in [7] is the following.…”

Section: Introductionmentioning

confidence: 99%

Approximate triangulations of Grassmann manifolds

Knudson

2020

Preprint

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Estimates of covering type and minimal triangulations based on category weight

2022

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In a recent publication [D. Govc, W. A. Marzantowicz and P. Pavešić, Estimates of covering type and the number of vertices of minimal triangulations, Discrete Comput. Geom. 63 2020, 1, 31–48], we have introduced a new method, based on the Lusternik–Schnirelmann category and the cohomology ring of a space X, that yields lower bounds for the size of a triangulation of X. In this current paper, we present an important extension that takes into account the fundamental group of X. In fact, if π 1 ⁢ ( X ) {\pi_{1}(X)} contains elements of finite order, then one can often find cohomology classes of high ‘category weight’, which in turn allow for much stronger estimates of the size of triangulations of X. We develop several weighted estimates and then apply our method to compute explicit lower bounds for the size of triangulations of orbit spaces of cyclic group actions on a variety of spaces including products of spheres, Stiefel manifolds, Lie groups and highly-connected manifolds.

show abstract

On the number of simplices required to triangulate a Lie group

Cited by 4 publications

References 17 publications

Estimates of covering type and minimal triangulations based on category weight

Estimates of covering type and minimal triangulations based on category weight

Approximate triangulations of Grassmann manifolds

Estimates of covering type and minimal triangulations based on category weight

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