The rational classification of links of codimension &gt; 2

Crowley, Diarmuid; Ferry, Steven C.; Skopenkov, Mikhail

doi:10.1515/form.2011.158

Cited by 5 publications

(7 citation statements)

References 21 publications

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“…This paper concludes the series of papers [2,3,4]. It is independent of previous ones in the sense that it uses statements from [4] but neither definitions nor methodology from any of them.…”

Section: Introductionmentioning

confidence: 78%

“…In Theorem 1.4 the inequality m < p + 3q/2 + 2 is assumed by aesthetic reasons -to reduce the number of cases and thus to simplify the statement and the proof. The classification of knotted tori for m ≥ p + 3q/2 + 2 is easier and is given by [20, Corollary 1.5], [23,Theorem 1.2], [4,Lemma 1.12].…”

Section: Knotted Torimentioning

confidence: 99%

“…For a finitely generated Abelian group G identify G ⊗ Q with Q rk G . We are going to use tacitly the following isomorphisms (see [10,Theorem 2.4] and [4,Theorem 1.13] respectively):…”

Section: Applicationsmentioning

confidence: 99%

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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: Introductionmentioning

confidence: 78%

Section: Knotted Torimentioning

confidence: 99%

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When is the set of embeddings finite up to isotopy?

Skopenkov

2015

Int. J. Math.

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“…Classification results for 2m < 3n + 4 concern links [1,2,12], embeddings of d-connected n-manifolds for 2m 3n + 3 − d [23,24], embeddings of 3-and 4dimensional manifolds [6-8, 28, 30], and rational classification of embeddings S p × S q → R m under stronger dimension restriction than m 2p + q + 3 [4,5] (see footnote 2). The methods of those papers essentially use the restrictions present there.…”

Section: Some General Motivationsmentioning

confidence: 99%

“…Group structures on sets of embeddings are constructed. 2 Then such relations are formulated in terms of exact sequences. The most non-trivial exact sequence is relation of knotted tori to links and knotted strips D p × S q → S m , i.e.…”

Section: Some General Motivationsmentioning

confidence: 99%

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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Rational homology and homotopy of high-dimensional string links

Tsopméné

Turchin

2018

Forum Mathematicum

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Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of {\pi_{0}}.

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The rational classification of links of codimension > 2

Cited by 5 publications

References 21 publications

When is the set of embeddings finite up to isotopy?

When is the set of embeddings finite up to isotopy?

Classification of knotted tori

Rational homology and homotopy of high-dimensional string links

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