How many simplices are needed to triangulate a Grassmannian?

Govc, Dejan; Marzantowicz, Wacław; Pavešić, Petar

doi:10.12775/tmna.2020.027

Cited by 9 publications

(14 citation statements)

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“…He used them to give what are still the best known estimates of the LS-category of real Grassmannians. The same products were also used in [9] to estimate the minimal size of triangulations of real Grassmannians. For the convenience of the reader we first summarize Stong's results and then derive further non-trivial cohomology products that will lead to new and much stronger estimates of topological complexity.…”

Section: Review Of Finite Grassmannians and Their Cohomologymentioning

confidence: 99%

Topological complexity of real Grassmannians

Pavešić¹

2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We use some detailed knowledge of the cohomology ring of real Grassmann manifolds G k (ℝ n ) to compute zero-divisor cup-length and estimate topological complexity of motion planning for k-linear subspaces in ℝ n . In addition, we obtain results about monotonicity of Lusternik–Schnirelmann category and topological complexity of G k (ℝ n ) as a function of n.

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Section: Review Of Finite Grassmannians and Their Cohomologymentioning

confidence: 99%

Topological complexity of real Grassmannians

Pavešić¹

2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…A strategy could consist in using the usual CW-structures of the Grassmannians, and converting them into simplicial complexes, as done theoretically in (Hatcher 2002, Theorem 2C.5). A recent result of Govc et al (2020) gives an idea about the complexity of this problem: the number of simplices of minimal triangulations of G d (R m ) must grow exponentially in both d and m.…”

Section: Discussionmentioning

confidence: 99%

Computing persistent Stiefel–Whitney classes of line bundles

Tinarrage

2021

J Appl. and Comput. Topology

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We propose a definition of persistent Stiefel-Whitney classes of vector bundle filtrations. It relies on seeing vector bundles as subsets of some Euclidean spaces. The usual Čech filtration of such a subset can be endowed with a vector bundle structure, that we call a Čech bundle filtration. We show that this construction is stable and consistent. When the dataset is a finite sample of a line bundle, we implement an effective algorithm to compute its first persistent Stiefel-Whitney class. In order to use simplicial approximation techniques in practice, we develop a notion of weak simplicial approximation. As a theoretical example, we give an in-depth study of the normal bundle of the circle, which reduces to understanding the persistent cohomology of the torus knot (1,2). We illustrate our method on several datasets inspired by image analysis.

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“…The characteristic classes are computed in a geometric way, as the algorithm uses explicit triangulations of the Grassmannians and pulls back the universal characteristic classes to a simplicial complex built from the sample. The practicality of the algorithms is limited by the fact that the number of simplices required to triangulate a Grassmannian Gr(d,n) is exponential in both d and n ( [18]), and the fact the algorithm often requires iterated subdivisions of the simplicial complex built from the data.…”

Section: Computation Of Characteristic Classesmentioning

confidence: 99%

Approximate and discrete Euclidean vector bundles

Scoccola¹,

Perea²

2021

Preprint

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We introduce ε-approximate versions of the notion of Euclidean vector bundle for ε ≥ 0, which recover the classical notion of Euclidean vector bundle when ε = 0. In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that ε-approximate vector bundles can be used to represent classical vector bundles when ε > 0 is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data, and also allows for some tolerance to noise when working with vector bundles in an applied setting.As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.

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How many simplices are needed to triangulate a Grassmannian?

Cited by 9 publications

References 0 publications

Topological complexity of real Grassmannians

Topological complexity of real Grassmannians

Computing persistent Stiefel–Whitney classes of line bundles

Approximate and discrete Euclidean vector bundles

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