On the Haefliger-Hirsch-Wu invariants for embeddings and immersions

Skopenkov, A.

doi:10.1007/s00014-002-8332-4

Cited by 27 publications

(26 citation statements)

References 54 publications

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“…E.g. by Theorem 2.9 we obtain that the Whitney invariant W : Emb 6k (S 2k−1 × S 2k ) → Z is surjective and for each u ∈ Z there is a 1-1 correspondence [Sk97,Sk02]. The embeddability in R 10 in Theorem 2.11.e is true also in the PL case, but this is covered by Theorem 2.3.b.…”

Section: The Whitney Obstructionmentioning

confidence: 92%

“…The Weber Theorem 8.1 can also be proved by first constructing the immersion and then modifying it to an embedding [Sk02].…”

Section: The Disjunction Methodsmentioning

confidence: 99%

“…The left squares are interesting in themselves and are useful elsewhere [Sk]. The definition of µ ′ here is simpler than that in [Sk02]. …”

Section: The Generalized Torus Lemma 62 [Sk02] (A) Ifmentioning

confidence: 99%

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Embedding and knotting of manifolds in Euclidean spaces

Skopenkov¹

2007

Surveys in Contemporary Mathematics

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Abstract. A clear understanding of topology of higher-dimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higher-dimensional topology in a way which makes clear the visual and intuitive part of the constructions and the arguments. In particular, we show how abstract algebraic constructions appear naturally in the study of geometric problems. Before giving a general construction, we illustrate the main ideas in simple but important particular cases, in which the essence is not veiled by technicalities.More specifically, we present several classical and modern results on the embedding and knotting of manifolds in Euclidean space. We state many concrete results (in particular, recent explicit classification of knotted tori). Their statements (but not proofs!) are simple and accessible to non-specialists. We outline a general approach to embeddings via the classical van Kampen-Shapiro-Wu-Haefliger-Weber 'deleted product' obstruction. This approach reduces the isotopy classification of embeddings to the homotopy classification of equivariant maps, and so implies the above concrete results. We describe the revival of interest in this beautiful branch of topology, by presenting new results in this area (of Freedman, Krushkal, Teichner, Segal, Spież and the author): a generalization the Haefliger-Weber embedding theorem below the metastable dimension range and examples showing that other analogues of this theorem are false outside the metastable dimension range.

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Section: The Whitney Obstructionmentioning

confidence: 92%

“…The Weber Theorem 8.1 can also be proved by first constructing the immersion and then modifying it to an embedding [Sk02].…”

Section: The Disjunction Methodsmentioning

confidence: 99%

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Embedding and knotting of manifolds in Euclidean spaces

Skopenkov¹

2007

Surveys in Contemporary Mathematics

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“…Данная темати-ка также заслуживает внимания благодаря многим интересным примерам (см. [6], [16], [17]). Ее систематическое исследование началось в работе [17].…”

Section: зацепления классификация зацеплений Sunclassified

Классификация Вложений Торов В 2-Метастабильной Размерности

Реповш¹,

Repovš²,

Skopenkov³

et al. 2012

Матем. сб.

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“…( 5 ) Almost embeddings and almost concordances were called quasi-embeddings and quasi-concordances in [Sk02].…”

Section: A Mapmentioning

confidence: 99%

A new invariant and parametric connected sum of embeddings

Skopenkov

2007

Fund. Math.

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Abstract.We define an isotopy invariant of embeddings N → R m of manifolds into Euclidean space. This invariant together with the α-invariant of Haefliger-Wu is complete in the dimension range where the α-invariant could be incomplete. We also define parametric connected sum of certain embeddings (analogous to surgery). This allows us to obtain new completeness results for the α-invariant and the following estimation of isotopy classes of embeddings. In the piecewise-linear category, for a (3n − 2m + 2)-connected nmanifold N with (4n + 5)/3 ≤ m ≤ (3n + 2)/2, each preimage of the α-invariant injects into a quotient of H 3n−2m+3 (N ), where the coefficients are Z for m − n odd and Z 2 for m − n even.

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On the Haefliger-Hirsch-Wu invariants for embeddings and immersions

Cited by 27 publications

References 54 publications

Embedding and knotting of manifolds in Euclidean spaces

Embedding and knotting of manifolds in Euclidean spaces

Классификация Вложений Торов В 2-Метастабильной Размерности

A new invariant and parametric connected sum of embeddings

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