Minimal m-functions

Hajduk, Boguslaw

doi:10.4064/fm-111-3-179-200

Cited by 6 publications

(9 citation statements)

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“…An interior critical point can be moved to the boundary and there split into two boundary critical points. A related result was stated in [Haj,Theorem 5]; we provide a rigorous proof under much weaker assumptions.…”

Section: Introductionmentioning

confidence: 61%

“…Elementary cancellation theorems. The following generalization of Milnor's theorem was first obtained in [Haj,Theorem 1].…”

Section: The Cancellation Of Boundary Handlesmentioning

confidence: 99%

“…Finally, in Section 5 we discuss the possibility of cancelling a pair of critical points. We include this part for completeness, the main result of this section was proved in [Haj,Theorem 1].…”

Section: Introductionmentioning

confidence: 99%

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Morse theory for manifolds with boundary

Borodzik

Nemethi

Ranicki

2016

Algebr. Geom. Topol.

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Abstract. We develop Morse theory for manifolds with boundary. Beside standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that under suitable connectedness assumptions a critical point in the interior of a Morse function can be moved to the boundary, where it splits into a pair of boundary critical points. As an application, we prove that every cobordism of connected manifolds with boundary splits as a union of left product cobordisms and right product cobordisms.

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Section: Introductionmentioning

confidence: 61%

“…Elementary cancellation theorems. The following generalization of Milnor's theorem was first obtained in [Haj,Theorem 1].…”

Section: The Cancellation Of Boundary Handlesmentioning

confidence: 99%

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Morse theory for manifolds with boundary

Borodzik

Nemethi

Ranicki

2016

Algebr. Geom. Topol.

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“…In this section, we recall some basic notions from Morse theory for manifolds with boundary from the well-written summary of [1]. The reader may also consult some of the original sources, such as [16,2,11]. We will assume that all manifolds (with or without boundaries or corners) are smooth.…”

Section: Cobordismsmentioning

confidence: 99%

Multiparameter persistent homology via generalized Morse theory

Bubenik¹,

Catanzaro²

2021

Preprint

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We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space. In the case of smooth functions on a compact manifold, we apply cobordism theory and Cerf theory to study the resulting persistence modules. We give examples in which we obtain a complete description of the persistence module as a direct sum of indecomposable summands and provide a corresponding visualization.

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“…Morse theory for manifolds with boundary was studied in the 1970s, by [Bra74] and [Haj81] independently. Recently, it was rediscovered by [KM07] in the context of Floer theory, and since then many articles about Morse theory for manifolds with boundary have appeared; see for instance [Lau11,Blo12,BNR12b].…”

Section: Introductionmentioning

confidence: 99%

Embedded Morse Theory and Relative Splitting of Cobordisms of Manifolds

Borodzik

Powell²

2014

J Geom Anal

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Abstract. We prove that an embedded cobordism between manifolds with boundary can be split into a sequence of right product and left product cobordisms, if the codimension of the embedding is at least two. This is a topological counterpart of the algebraic splitting theorem for embedded cobordisms of the first author, A. Némethi and A. Ranicki. In the codimension one case, we provide a slightly weaker statement.We also give proofs of rearrangement and cancellation theorems for handles of embedded submanifolds with boundary.

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Minimal m-functions

Cited by 6 publications

References 0 publications

Morse theory for manifolds with boundary

Morse theory for manifolds with boundary

Multiparameter persistent homology via generalized Morse theory

Embedded Morse Theory and Relative Splitting of Cobordisms of Manifolds

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