When is the set of embeddings finite up to isotopy?

Skopenkov, Mikhail

doi:10.1142/s0129167x15500512

Cited by 5 publications

(13 citation statements)

References 21 publications

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“…the νσ(iζλ )-sequence from the proof of Theorem 1.2 in §2.2. This is the main theoretical result [Sk15, Theorem 1.6] of [Sk15], which non-trivally extends [Sk06, Restriction Lemma 5.2] and Lemma 2.15.a (see footnote 8).…”

Section: Some General Motivationssupporting

confidence: 62%

“…The new ideas allowing to go beyond the above results follow [Sk15] and unpublished work [Sk06]. One idea is to find relations between different sets of (isotopy classes of) embeddings, invariants of embeddings and geometric constructions of embeddings.…”

Section: Some General Motivationsmentioning

confidence: 95%

“…Proof that σ λ = 0. 19 (This proof suggests an alternative definition of ζλ allowing minor simplification in [Sk15].) Take an embedding F : T p,q+1…”

Section: Appendix: New Direct Proof Of Corollary 15amentioning

confidence: 99%

“…Using the (m − q)-framing identify each fiber of the normal bundle to f (D q+1 − ) with the space R m−q . Define a map S q → V m−q,p+1 by mapping point x ∈ S q to the (p + 1)-framing at the point f (x [CRS12,Sk15].…”

Section: Calculationsmentioning

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: Some General Motivationssupporting

confidence: 62%

Section: Some General Motivationsmentioning

confidence: 95%

“…Proof that σ λ = 0. 19 (This proof suggests an alternative definition of ζλ allowing minor simplification in [Sk15].) Take an embedding F : T p,q+1…”

Section: Appendix: New Direct Proof Of Corollary 15amentioning

confidence: 99%

Section: Calculationsmentioning

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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“…This notion generalizes both links (l k = 0) and framed links (l k = m − p k ) [12]. Partially framed links play an important role in the classification of embeddings of general manifolds [3,23,26].…”

Section: Framed Linksmentioning

confidence: 98%

The rational classification of links of codimension > 2

Crowley¹,

Ferry

Skopenkov

2014

Forum Mathematicum

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Let m and p 1 ; : : : ; p r < m 2 be positive integers. The set of links of codimension > 2, E m .F r kD1 S p k /, is the set of smooth isotopy classes of smooth embeddings F r kD1 S p k ! S m . Haefliger showed that E m . F r kD1 S p k / is a finitely generated abelian group with respect to embedded connected summation and computed its rank in the case of knots, i.e. r D 1. For r > 1 and for restrictions on p 1 ; : : : ; p r the rank of this group can be computed using results of Haefliger or Nezhinsky. Our main result determines the rank of the group E m . F r kD1 S p k / in general. In particular we determine precisely when E m .F r kD1 S p k / is finite. We also accomplish these tasks for framed links. Our proofs are based on the Haefliger exact sequence for groups of links and the theory of Lie algebras.

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Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification

Crowley¹,

Skopenkov²

2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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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 $H_q := H_q(N; {\mathbb Z} )$ . Our main result is a readily calculable classification of embeddings $N \to {\mathbb R}^7$ up to isotopy, with an indeterminacy. Such a classification was only known before for $H_1=0$ by our earlier work from 2008. Our classification is complete when $H_2=0$ or when the signature of $N$ is divisible neither by 64 nor by 9. The group of knots $S^4\to {\mathbb R}^7$ acts on the set of embeddings $N\to {\mathbb R}^7$ up to isotopy by embedded connected sum. In Part I we classified the quotient of this action. The main novelty of this paper is the description of this action for $H_1 \ne 0$ , with an indeterminacy. Besides the invariants of Part I, detecting the action of knots involves a refinement of the Kreck invariant from our work of 2008. For $N=S^1\times S^3$ we give a geometrically defined 1–1 correspondence between the set of isotopy classes of embeddings and a certain explicitly defined quotient of the set ${\mathbb Z} \oplus {\mathbb Z} \oplus {\mathbb Z} _{12}$ .

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When is the set of embeddings finite up to isotopy?

Cited by 5 publications

References 21 publications

Classification of knotted tori

Classification of knotted tori

The rational classification of links of codimension > 2

Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification

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