Discrete Morse Theory for Computing Cellular Sheaf Cohomology

Curry, Justin; Nanda, Vidit

doi:10.1007/s10208-015-9266-8

Cited by 42 publications

(50 citation statements)

References 53 publications

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“…By abuse of notation, we refer to the simplices by their labeling under the discrete Morse function (the fact that pairs in V f are given the same label should not cause confusion). For each of the pairs (9,9), (6,6), (3,3), (2, 2), (1, 1) ∈ V f , we have the corresponding values l 9 = 11, l 6 = 8, l 3 = 5, l 2 = 2, and l 1 = 1. The corresponding strong collapses under the indicated vertices are given by S f 9 = {(9, 9), (10, 10), (11,11)}, S Hence there is a single critical pair, namely, (13,13), so that scrit(f ) = {0, (13, 13)}.…”

Section: Strong Discrete Morse Theorymentioning

confidence: 99%

“…Since its inception by Robin Forman [9], discrete Morse theory has been a powerful and versatile tool used not only in diverse fields of mathematics, but also in applications to other areas [14] as well as a computational tool [6]. Its adaptability stems in part from the fact that it is a discrete version of the beautiful and successful "smooth" Morse theory [11].…”

Section: Introductionmentioning

confidence: 99%

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Strong Discrete Morse Theory and Simplicial L–S Category: A Discrete Version of the Lusternik–Schnirelmann Theorem

Fernández-Ternero

Macías–Virgós

Scoville

et al. 2019

Discrete Comput Geom

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We prove a discrete version of the Lusternik-Schnirelmann theorem for discrete Morse functions and the recently introduced simplicial Lusternik-Schnirelmann category of a simplicial complex. To accomplish this, a new notion of critical object of a discrete Morse function is presented, which generalizes the usual concept of critical simplex (in the sense of R. Forman). We show that the non-existence of such critical objects guarantees the strong homotopy equivalence (in the Barmak and Minian's sense) between the corresponding sublevel complexes. Finally, we establish that the number of critical objects of a discrete Morse function defined on K is an upper bound for the non-normalized simplicial Lusternik-Schnirelmann category of K.

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Section: Strong Discrete Morse Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Strong Discrete Morse Theory and Simplicial L–S Category: A Discrete Version of the Lusternik–Schnirelmann Theorem

Fernández-Ternero

Macías–Virgós

Scoville

et al. 2019

Discrete Comput Geom

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“…This approach has the double advantage of reducing both time and memory complexity. This goal has successfully been reached by the use of discrete Morse theory [22,25,34] (see also [19,26]), and led to the implementation of the efficient software Perseus [36].…”

Section: Introductionmentioning

confidence: 99%

Discrete Morse Theory for Computing Zigzag Persistence

Maria

Schreiber

2019

Lecture Notes in Computer Science

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We introduce a theoretical and computational framework to use discrete Morse theory as an efficient preprocessing in order to compute zigzag persistent homology.From a zigzag filtration of complexes (Xi), we introduce a zigzag Morse filtration whose complexes (Ai) are Morse reductions of the original complexes (Xi), and we prove that they both have same persistent homology. This zigzag Morse filtration generalizes the filtered Morse complex of Mischaikow and Nanda [34], defined for standard persistence.The maps in the zigzag Morse filtration are forward and backward inclusions, as is standard in zigzag persistence, as well as a new type of map inducing non trivial changes in the boundary operator of the Morse complex. We study in details this last map, and design algorithms to compute the update both at the complex level and at the homology matrix level when computing zigzag persistence. We deduce an algorithm to compute the zigzag persistence of a filtration that depends mostly on the number of critical cells of the complexes, and show experimentally that it performs better in practice.

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“…Forman's discrete Morse theory [14] has been successfully used to perform (co)homology computations not only in algebraic topology [11,30,31], but also in commutative algebra [21], topological combinatorics [37], algebraic combinatorics [35] and even geometric group theory [4]. The central idea involves the imposition of a partial matching µ on adjacent cell pairs of a regular CW complex X subject to a global acyclicity condition -the unmatched cells play the role of critical points whereas the matched cells generate combinatorial gradient-like flow paths.…”

Section: Introductionmentioning

confidence: 99%

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 Morse Theory for Computing Cellular Sheaf Cohomology

Cited by 42 publications

References 53 publications

Strong Discrete Morse Theory and Simplicial L–S Category: A Discrete Version of the Lusternik–Schnirelmann Theorem

Strong Discrete Morse Theory and Simplicial L–S Category: A Discrete Version of the Lusternik–Schnirelmann Theorem

Discrete Morse Theory for Computing Zigzag Persistence

Discrete Morse theory and localization

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