Link homotopy in the 2-metastable range

Habegger, Nathan; Kaiser, Uwe

doi:10.1016/s0040-9383(97)00010-4

Cited by 24 publications

(33 citation statements)

References 16 publications

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“…So r-fold analogue of Haefliger's h-principle for almost r-embeddings and of the Whitney trick is an important contribution of Mabillard and Wagner. Their r-fold Whitney trick involves analogue of increasing the connectivity (surgery) of the intersection set [Ha63], [HK,Theorem 4.5 and appendix A], [CRS,Theorem 4.7 and appendix]. In other words, this is first attaching an embedded 1-handle along an arc ('piping') and then attaching a canceling embedded 2-handle along a disk ('unpiping') [Ha62,§3], [Me, proof of Theorem 1.1 in p. 7].…”

Section: On References Concerning Theorem 12mentioning

confidence: 99%

A user's guide to the topological Tverberg conjecture

Skopenkov¹

2018

Russ. Math. Surv.

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The well-known topological Tverberg conjecture was considered a central unsolved problem of topological combinatorics. The conjecture asserts that for each integers r, d > 1 and each continuous map f :A proof for a prime power r was given by I. Bárány, S. Shlosman, A. Szűcs, M. Özaydin and A. Volovikov in 1981-1996. A counterexample for other r was found in a series of papers most of them recent. (The exact description of contribution of particular authors is more complex and we provide more details in historical notes.)The arguments form a beautiful and fruitful interplay between combinatorics, algebra and topology. In this expository note we present a simplified explanation of easier parts of the arguments, accessible to non-specialists in the area.1 The exact description of contribution of particular authors is more complex and we provide more details in §3.2. We do not claim that the contributions were equal, but leave it to a reader to make his/her own opinion. M. de Longueville is not included in this list because Lemma 3.1 is used not to produce a counterexample, but to construct counterexamples for d > 3r + 1 from a counterexample for d = 3r + 1.

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Section: On References Concerning Theorem 12mentioning

confidence: 99%

A user's guide to the topological Tverberg conjecture

Skopenkov¹

2018

Russ. Math. Surv.

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“…Это наиболее трудный шаг доказательства, который использует предположения β(f ) = 0 и m p + 4 3 q + 2. В силу результатов Хабеггера-Кайзера из работы [30], можно считать, что сфе-роид f | * ×S q стягиваем вне f B p+q . Таким образом, мы можем заклеить сфероид f ( * × S q ) (не обязательно вложенным) диском D q+1 , расположенным в про-странстве S m − f B p+q .…”

Section: почти вложенияunclassified

“…Это необходимо для доказательства леммы 1 о до-полнении, сформулированной в § 4. Наше изложение полностью аналогично приложению A из [30], но производится в большей общности и более подробно.…”

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Классификация Вложений Торов В 2-Метастабильной Размерности

Реповш¹,

Repovš²,

Skopenkov³

et al. 2012

Матем. сб.

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“…For embeddings up to homotopy see [Co69,St,Wa70, §11, Hu70', CW78,Ha84]. For the classification of link maps see [Mi54,MR86,Ko88,Ko90,Ma90,HK98,Sk00]. For embeddings of polyhedra in some manifolds see [Wa66,Ne68,LS69,RBS99].…”

Section: Definitions and Notationsmentioning

confidence: 99%

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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Link homotopy in the 2-metastable range

Cited by 24 publications

References 16 publications

A user's guide to the topological Tverberg conjecture

A user's guide to the topological Tverberg conjecture

Классификация Вложений Торов В 2-Метастабильной Размерности

Embedding and knotting of manifolds in Euclidean spaces

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