How do autodiffeomorphisms act on embeddings?

Skopenkov, A.

doi:10.1017/s030821051700021x

Cited by 2 publications

(2 citation statements)

References 31 publications

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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: 92%

“…Classification of knotted tori is a natural next step after the Haefliger link theory [12] and the classification of embeddings of highly-connected manifolds [27, § 2], [15]. Such a step gives some insight or even precise information concerning embeddings of arbitrary manifolds [26,31,33], and reveals new interesting relations to algebraic topology.…”

Section: Some General Motivationsmentioning

confidence: 99%

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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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Section: Definitions Of [·]mentioning

confidence: 92%

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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Embeddings of non-simply-connected 4-manifolds in 7-space. I. Classification modulo knots

Crowley¹,

Skopenkov²

2016

Preprint

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We work in the smooth category. Let N be a closed connected orientable 4-manifold with torsion free H 1 , where H q := H q (N ; Z). The main result is a complete readily calculable classification of embeddings N → R 7 , up to equivalence which is isotopy and embedded connected sum with embeddings S 4 → R 7 . Such a classification was earlier known only for H 1 = 0 by Boéchat-Haefliger-Hudson 1970. Our classification involves Boéchat-Haefliger invariant κ(f ) ∈ H 2 , Seifert bilinear form λ(f ) : H 3 × H 3 → Z and β-invariant assuming values in the quotient of H 1 defined by values of κ(f ) and λ(f ).In particular, for N = S 1 × S 3 we define geometrically a 1-1 correspondence between the set of equivalence classes of embeddings and an explicitly defined quotient of Z ⊕ Z.Our proof is based on Kreck modified surgery approach, and also uses parametric connected sum.2 This is proved analogously to the case X = D 0 + of [Sk15, Standardization Lemma 2.1.b], cf. [Sk15, Remark 2.3.a], because the construction of # has an analogue for isotopy.

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How do autodiffeomorphisms act on embeddings?

Cited by 2 publications

References 31 publications

Classification of knotted tori

Classification of knotted tori

Embeddings of non-simply-connected 4-manifolds in 7-space. I. Classification modulo knots

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