A new invariant and parametric connected sum of embeddings

Skopenkov, A.

doi:10.4064/fm197-0-12

Cited by 14 publications

(21 citation statements)

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“…A natural next step (after link theory and the classification of embeddings of highly-connected manifolds) towards classification of embeddings of arbitrary manifolds is the classification of knotted tori, i.e., embeddings S p × S q → S m . The classification of knotted tori gives some insight or even precise information concerning arbitrary manifolds [21]; see also Theorem 4.1 below. Many interesting examples of embeddings are knotted tori [13,18,20,15].…”

Section: Knotted Torimentioning

confidence: 99%

When is the set of embeddings finite up to isotopy?

Skopenkov

2015

Int. J. Math.

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Given a manifold N and a number m, we study the following question: is the set of isotopy classes of embeddings N → S m finite? In case when the manifold N is a sphere the answer was given by A. Haefliger in 1966. In case when the manifold N is a disjoint union of spheres the answer was given by D. Crowley, S. Ferry and the author in 2011. We consider the next natural case when N is a product of two spheres. In the following theorem, F CS(i, j) ⊂ Z 2 is a specific set depending only on the parity of i and j which is defined in the paper.Theorem. Assume that m > 2p + q + 2 and m < p + 3q/2 + 2. Then the set of C 1 -isotopy classes of C 1 -smooth embeddings S p × S q → S m is infinite if and only if either q + 1 or p + q + 1 is divisible by 4, or there exists a point (x, y) Our approach is based on a group structure on the set of embeddings and a new exact sequence, which in some sense reduces the classification of embeddings S p × S q → S m to the classification of embeddings S p+q ⊔ S q → S m and D p × S q → S m . The latter classification problems are reduced to homotopy ones, which are solved rationally.

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Section: Knotted Torimentioning

confidence: 99%

When is the set of embeddings finite up to isotopy?

Skopenkov

2015

Int. J. Math.

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“…The sum operation on E m (S p × S q ) is 'S p -parametric connected sum', cf. [Sk07,Sk10', MAP], [Sk17,Theorem 8]. See Group Structure Lemma 2.2, Remark 2.3 on comparison to previous work and Remark 2.4 on the dimension restrictions.…”

Section: Definitions Of [·]mentioning

confidence: 94%

Classification of knotted tori

Skopenkov

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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For a smooth manifold N denote by E m (N ) the set of smooth isotopy classes of smooth embeddings N → R m . A description of the set E m (S p × S q ) was known only for p = q = 0 or for p = 0, m = q + 2 or for 2m ≥ 2(p + q) + max{p, q} + 4 (in terms of homotopy groups of spheres and Stiefel manifolds). For m ≥ 2p + q + 3 an abelian group structure on E m (S p × S q ) is introduced. We prove that this group andare 'isomorphic up to an extension problem'. Here λ U : E → π q (S m−p−q−1 ) is the linking coefficient defined on the subset E ⊂ E m (S q S p+q ) formed by isotopy classes of embeddings whose restriction to each component is unknotted.This result and its proof have corollaries which, under stronger dimension restrictions, more explicitly describe E m (S p × S q ) in terms of homotopy groups of spheres and Stiefel manifolds. The proof is based on relations between sets E m (N ) for different N and m, in particular, on a recent exact sequence of M. Skopenkov.

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“…Рассмот-рение заузленных торов -следующий естественный шаг после исследования зацеплений по направлению к классификации вложений произвольных много-образий (см. [14], [15]) согласно теореме о разбиении на ручки. Данная темати-ка также заслуживает внимания благодаря многим интересным примерам (см.…”

Section: зацепления классификация зацеплений Sunclassified