Surfaces in 4-Manifolds: Concordance, Isotopy, and Surgery

Sunukjian, Nathan

doi:10.1093/imrn/rnu187

Cited by 28 publications

(49 citation statements)

References 41 publications

(27 reference statements)

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“…Proof. We sketch the standard argument for (a); compare [44,49,64]. Since S and T are embedded in C + with simply connected complements, they are homologous, from which it follows (using Freedman [33]) that there is a self-homeomorphism of X + taking S to T .…”

Section: Definitionmentioning

confidence: 99%

Stable isotopy in four dimensions

Auckly

Kim

Melvin

et al. 2015

Journal of the London Mathematical Society

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We construct infinite families of topologically isotopic, but smoothly distinct knotted spheres, in many simply connected 4-manifolds that become smoothly isotopic after stabilizing by connected summing with S 2 ×S 2 , and as a consequence, analogous families of diffeomorphisms and metrics of positive scalar curvature for such 4-manifolds. We also construct families of smoothly distinct links, all of whose corresponding proper sublinks are smoothly isotopic, that become smoothly isotopic after stabilizing.

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Section: Definitionmentioning

confidence: 99%

Stable isotopy in four dimensions

Auckly

Kim

Melvin

et al. 2015

Journal of the London Mathematical Society

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“…On the other hand, many examples of exotic embeddings in the literature, such as the examples of Σ i in S 4 constructed in [9,10], can be seen to be topologically isotopic [29], and are thus exotically knotted.…”

Section: Introductionmentioning

confidence: 99%

“…Not all exotic embeddings are exotic knottings; one can, for instance, have ambiently homeomorphic Σ i representing different homology classes [4]. On the other hand, many examples of exotic embeddings in the literature, such as the examples of Σ i in S 4 constructed in [9,10], can be seen to be topologically isotopic [29], and are thus exotically knotted.…”

Section: Introductionmentioning

confidence: 99%

Knotted surfaces in 4-manifolds and stabilizations

Baykur

Sunukjian

2015

J Topology

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Abstract. In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4-manifold, which can moreover assumed to be attached in a standard way (locally and unknottedly) in many favorable situations. In particular, any exotically knotted pair of surfaces with cyclic fundamental group complements become smoothly isotopic after a same number of standard stabilizations -analogous to C.T.C. Wall's celebrated result on the stable equivalence of simply-connected 4-manifolds. We moreover show that all constructions of exotic knottings of surfaces we are aware of, which display a good variety of techniques and ideas, produce surfaces that become smoothly isotopic after a single stabilization.

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“…Since the spheres

A

and

B

are homologous, this map acts like the identity on

H_{2} false(X^{\circ} false)

, and therefore the spheres

A

and

B

are equivalent. Since

A

and

B

are homologous and have simply connected complement in

X^{\circ}

, they must also be topologically isotopic by Lee and Wilczyński (and more recently Sunukjian ). However,

A

and

B

are not smoothly isotopic in

X^{\circ}

.…”

Section: Main Examplesmentioning

confidence: 99%

“…Since A and B are homologous and have simply connected complement in X • , they must also be topologically isotopic by Lee and Wilczyński [9] (and more recently Sunukjian [20]). However, A and B are not smoothly isotopic in X • .…”

Section: Main Examplesmentioning

confidence: 99%

Equivalent non‐isotopic spheres in 4‐manifolds

Schwartz

2019

Journal of Topology

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We construct infinitely many smooth oriented 4‐manifolds containing pairs of homotopic, smoothly embedded 2‐spheres that are not topologically isotopic, but that are equivalent by an ambient diffeomorphism inducing the identity on homology. These examples show that Gabai's recent ‘generalized 4D lightbulb theorem' does not generalize to arbitrary 4‐manifolds. In contrast, we also show that there are smoothly embedded 2‐spheres that are both equivalent and topologically isotopic, but not smoothly isotopic.

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Surfaces in 4-Manifolds: Concordance, Isotopy, and Surgery

Cited by 28 publications

References 41 publications

Stable isotopy in four dimensions

Stable isotopy in four dimensions

Knotted surfaces in 4-manifolds and stabilizations

Equivalent non‐isotopic spheres in 4‐manifolds

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