Discrete gradient fields on infinite complexes

Vilches, José Antonio; Jerše, Gregor; Kosta, Neźa Mramor

doi:10.3934/dcds.2011.30.623

Cited by 5 publications

(8 citation statements)

References 14 publications

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“…It is thus natural to address the homotopy properties of the complexes D n,k in (1) starting with the (unknown) instances having small values of k. Our main result in this paper, Theorem 1.3 below, addresses the homotopy properties of D n,n−2 , the first step in the task we just set forth.…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

“…The D n,k analogue of the former generators (the 4-cycles) were identified by Vizing: Theorem 1.1 (Vizing [5]). Let 2 ≤ k ≤ n. The dimension of the complex D n,k is one less than the integral part of 1 2…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

“…The first indication that D 2 should lead to a suitable homotopy model for D 2 comes from the results in [1,4] and the fact that D 2 is rayless. In fact, it follows from the proof of Proposition 4.1 that there are no infinite paths in D 2 .…”

Section: Graphs With Vertices In the Natural Numbersmentioning

confidence: 99%

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On the homotopy type of complexes of graphs with bounded domination number

González,

Hoekstra-Mendoza

2019

Preprint

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Let D n,γ be the complex of graphs on n vertices and domination number at least γ. We prove that D n,n−2 has the homotopy type of a finite wedge of 2-spheres. This is done by using discrete Morse theory techniques. Acyclicity of the needed matching is proved by introducing a relativized form of a much used method for constructing acyclic matchings on suitable chunks of simplices. Our approach allows us to extend our results to the realm of infinite graphs. We give evidence supporting the assertion that the homotopy equivalence D n,n−2 ≃ Nn S 2 does not generalize as expected for D n,γ , if γ ≤ n − 3.

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Section: Preliminaries and Main Resultsmentioning

confidence: 99%

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

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On the homotopy type of complexes of graphs with bounded domination number

González,

Hoekstra-Mendoza

2019

Preprint

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“…There is a small gap in the proof of Gallais's theorem, but this has recently been fixed by Benedetti [Benedetti (2014)], who also proved a stronger version of the result.…”

Section: Open Questions and Future Workmentioning

confidence: 99%

“…Forman proved Theorem 2.7 only for finite cell complexes. It is also true for infinite cell complexes; see [Ayala, et al (2011)]. …”

Section: Theorem 23 ([mentioning

confidence: 99%

Birth and death in discrete Morse theory

King

Knudson

Kosta

2017

Journal of Symbolic Computation

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Abstract. Suppose M is a finite cell decomposition of a space X and that for 0 = t 0 < t 1 < · · · < tr = 1 we have a discrete Morse function Ft i : M → R. In this paper, we study the births and deaths of critical cells for the functions Ft i and present an algorithm for pairing the cells that occur in adjacent slices. We first study the case where the cell decomposition of X is the same for each t i , and then generalize to the case where they may differ. This has potential applications in topological data analysis, where one has function values at a sample of points in some region in space at several different times or at different levels in an object.

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Discrete Morse theory and localization

Nanda¹

2019

Journal of Pure and Applied Algebra

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Incidence relations among the cells of a regular CW complex produce a posetenriched category of entrance paths whose classifying space is homotopy-equivalent to that complex. We show here that each acyclic partial matching (in the sense of discrete Morse theory) of the cells corresponds precisely to a homotopy-preserving localization of the associated entrance path category. Restricting attention further to the full localized subcategory spanned by critical cells, we obtain the discrete flow category whose classifying space is also shown to lie in the homotopy class of the original CW complex. This flow category forms a combinatorial and computable counterpart to the one described by Cohen, Jones and Segal in the context of smooth Morse theory.

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Discrete gradient fields on infinite complexes

Cited by 5 publications

References 14 publications

On the homotopy type of complexes of graphs with bounded domination number

On the homotopy type of complexes of graphs with bounded domination number

Birth and death in discrete Morse theory

Discrete Morse theory and localization

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