Symmetric Whitney tower cobordism for bordered 3-manifolds and links

Cha, Jae Choon

doi:10.48550/arxiv.1204.4968

Cited by 2 publications

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“…In this paper we show, in a strong sense involving homotopy of meridians, that the answer is negative for a large class of links satisfying a certain nonvanishing condition on Milnor's µ-invariants, even in the framework of symmetric grope and Whitney tower generalisations of concordance and homology cobordism in the sense of [COT03,Cha]. Also we employ topological surgery in dimension 4 to give a new construction of homology cobordisms of zero surgery manifolds.…”

Section: Introductionmentioning

confidence: 92%

“…There are many linking number one 2-component links which are not concordant, as can be detected, for example, by the multivariable Alexander polynomial [Kaw78,Nak78]. For related in-depth study, the reader is referred to, for instance, [CK99,FP,Cha]. With our respective coauthors, we gave non-concordant linking number one links with two unknotted components, for which abelian invariants such as the multivariable Alexander polynomial are unable to obstruct them from being concordant.…”

Section: Link Concordance Versus Zero Surgery Homology Cobordismmentioning

confidence: 99%

“…Our results serve to underline the philosophy that when investigating the relative problem of whether two links are concordant, and neither of them are the unlink, one should consider obstructions to homology cobordism of the link exteriors viewed as bordered manifolds, rather than to homology cobordism of the zero surgery manifolds, even in low dimensions. This was implemented in, for example [Kaw78,Nak78,Cha] (see also [FP] for a related approach).…”

Section: Link Exteriors and The Homology Surgery Approachmentioning

confidence: 99%

“…To construct links which are grope concordant, we employ the method of [Cha,Section 4]. We begin by giving the definition of grope concordance.…”

Section: Satellite Construction and Grope Concordance Of Linksmentioning

confidence: 99%

“…Following [Cha,Definition 4.2], we call (L, η) a satellite configuration of height k if L is a link in S 3 , η is an unknotted circle in S 3 disjoint from L, and the 0-linking parallel of η in…”

Section: Definition 41 ([Ft95]mentioning

confidence: 99%

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Nonconcordant links with homology cobordant zero-framed surgery manifolds

Choon

Powell

2014

Pacific J. Math.

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. NONCONCORDANT LINKS WITH HOMOLOGY COBORDANT ZERO-FRAMED SURGERY MANIFOLDS JAE CHOON CHA AND MARK POWELLWe use topological surgery theory to give sufficient conditions for the zeroframed surgery manifold of a 3-component link to be homology cobordant to the zero-framed surgery on the Borromean rings (also known as the 3-torus) via a topological homology cobordism preserving the free homotopy classes of the meridians. This enables us to give examples of 3-component links with unknotted components and vanishing pairwise linking numbers, such that any two of these links have homology cobordant zero-surgeries in the above sense, but the zero-surgery manifolds are not homeomorphic. Moreover, the links are not concordant to one another, and in fact they can be chosen to be height h but not height h + 1 symmetric grope concordant, for each h which is at least three.

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Section: Introductionmentioning

confidence: 92%

Section: Link Concordance Versus Zero Surgery Homology Cobordismmentioning

confidence: 99%

Section: Link Exteriors and The Homology Surgery Approachmentioning

confidence: 99%

“…To construct links which are grope concordant, we employ the method of [Cha,Section 4]. We begin by giving the definition of grope concordance.…”

Section: Satellite Construction and Grope Concordance Of Linksmentioning

confidence: 99%

Section: Definition 41 ([Ft95]mentioning

confidence: 99%

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Nonconcordant links with homology cobordant zero-framed surgery manifolds

Choon

Powell

2014

Pacific J. Math.

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Links not concordant to the Hopf link

Friedl¹,

Powell²

2014

Math. Proc. Camb. Phil. Soc.

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We give new Casson-Gordon style obstructions for a two-component link to be topologically concordant to the Hopf link.1 Lemma 3.2. Let p, q be primes. For any two-component link J withProof. Since the linking number is 1, the inclusion maps ∂X 1 J → X J are Z-homology equivalences. We wish to apply Proposition 3.1 with S = S 1 × S 1 to see that H * (X ϕ J ; Q(ξ q k )(H)) ∼ = H * ((∂X 1 J ) ϕ ; Q(ξ q k )(H)) ∼ = 0.

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Symmetric Whitney tower cobordism for bordered 3-manifolds and links

Cited by 2 publications

References 0 publications

Nonconcordant links with homology cobordant zero-framed surgery manifolds

Nonconcordant links with homology cobordant zero-framed surgery manifolds

Links not concordant to the Hopf link

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