Chern–Simons functional and the homology cobordism group

Daemi, Aliakbar

doi:10.1215/00127094-2020-0017

Cited by 23 publications

(25 citation statements)

References 37 publications

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“…In the case that L is a knot, the question was answered affirmatively by the first named author, under the assumption that X admits a handle structure with no 3-handles [7, Theorem [8] that every link in a homology sphere is homology concordant to a link in S 3 . This conjecture is particularly intriguing because the corresponding statement is known to be false in the smooth category [16,15,25,6]. Evidence for this conjecture was provided in [7,8] and we provide a similar type of evidence in this article.…”

Section: Introductionsupporting

confidence: 73%

The relative Whitney trick and its applications

Davis¹,

Orson²,

Park³

2021

Preprint

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We introduce a geometric operation, which we call the relative Whitney trick, that removes a single double point between properly immersed surfaces in a 4-manifold with boundary. Using the relative Whitney trick we prove that every link in a homology sphere is homotopic to a link that is topologically slice in a contractible topological 4-manifold. We further prove that links in homology spheres are equivalent to links in S 3 up to any level of the Whitney tower filtration that approximates link concordance. Finally, we explore the minimum Gordian distance from a link in S 3 to a homotopically trivial link. Extending this notion to links in homology spheres, we use the relative Whitney trick to make explicit computations for 3-component links and establish bounds in general.

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Section: Introductionsupporting

confidence: 73%

The relative Whitney trick and its applications

Davis¹,

Orson²,

Park³

2021

Preprint

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“…Secondly, if K is homotopic in Y to a knot which bounds an embedded disk in a homology ball, then K is nullhomotopic in that homology ball. By work of Daemi [4,Remark 1.6] there exist a knot K in a homology sphere Y such that Y bounds a homology ball and yet K is not nullhomotopic in any homology ball bounded by Y . Such a knot cannot be homotopic to a knot which bounds a smoothly embedded disk.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…If one allows homology spheres which bound homology balls but not contractible 4manifolds then there is an obstruction to a knot being related to a homology slice knot by a sequence of homotopies and homology concordances. Indeed, by [4,Remark 1.6] there exist knots in homology spheres which are not nullhomotopic in any homology ball. Such a knot cannot be reduced to a smoothly slice knot by any sequence of homotopies and homology concordances.…”

Section: Said Another Way If We Letmentioning

confidence: 99%

“…There is an extension of the equivalence relations 3 h and c . Given knots K and J in homology spheres Y and X, if there exists a smooth homology cobordism from Y to X in which K and J are homotopic then we write (Y, K) 4 h (X, J). Equivalently and more geometrically we say (Y, K) 4 h (X, J) if there exists a smooth homology cobordism from Y to X in which K and J cobound an immersed annulus.…”

Section: Said Another Way If We Letmentioning

confidence: 99%

“…Given knots K and J in homology spheres Y and X, if there exists a smooth homology cobordism from Y to X in which K and J are homotopic then we write (Y, K) 4 h (X, J). Equivalently and more geometrically we say (Y, K) 4 h (X, J) if there exists a smooth homology cobordism from Y to X in which K and J cobound an immersed annulus. It is clear that if (Y, K) and (X, J) are related by a sequence of homotopies and homology concordances then they are related under 4 h .…”

Section: Said Another Way If We Letmentioning

confidence: 99%

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Concordance, Crossing Changes, and Knots in Homology Spheres

Davis¹

2019

Can. Math. Bull.

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Any knot in S 3 may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that knot is smoothly concordant to a knot which is homotopic to a smoothly slice knot. As a consequence, we prove that the equivalence relation on knots in homology spheres given by cobounding immersed annuli in a homology cobordism is generated by concordance in homology cobordisms together with homotopy in a homology sphere.

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