The Self-Intersections of a Smooth n-Manifold in 2n-Space

Whitney, Hassler

doi:10.1007/978-1-4612-2974-2_5

Cited by 112 publications

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“…We have counted crossings along edges modulo 2, corresponding to the traditional Hanani-Tutte theorem; however, there is another way to count crossings that might be worth exploring, and that goes back at least as far as Whitney's 1944 paper [55], though Flores already hints at the possibility [20, 62. Kolloquium §12]; it's first explicitly worked out by Tutte.…”

Section: Algebraic Characterizations Of Planaritymentioning

confidence: 99%

“…On the question of whether iacr = cr, where cr is the traditional crossing number (see Section 3.4), Levow [17] writes "it seems reasonable to hope that equality holds for all graphs"; as we will see in Section 3.4 this is not the case, the two notions of crossing numbers differ. Interestingly, Whitney came close to asking the same question 30 years earlier [55]. Levow continues "whether or not equality holds, the algebraic setting may be useful in helping to compute crossing numbers, for it leads to a lower bound for the crossing number given in terms of the solution to an integer or Boolean minimization problem.…”

Section: Theorem 119 (Tutte) A(g) Is a Coset Ofmentioning

confidence: 99%

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Hanani-Tutte and Related Results

Schaefer

2013

Bolyai Society Mathematical Studies

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We investigate under what conditions crossings of adjacent edges and pairs of edges crossing an even number of times are unnecessary when drawing graphs. This leads us to explore the Hanani-Tutte theorem and its close relatives, emphasizing the intuitive geometric content of these results. The Hanani-Tutte Theorem in The PlaneIn 1934, Hanani [15] published a paper which-in passing-established the following result:Any drawing of a K 5 or a K 3,3 contains two independent edges crossing each other oddly. 1Since by Kuratowski's theorem every non-planar graph contains a subdivision of K 5 or K 3,3 , Hanani's observation implies that any drawing of a nonplanar graph contains two vertex-disjoint paths that cross an odd number of times and therefore two independent edges that cross oddly-one in each path. This consequence was first explicitly stated by Tutte [49]. 2 1 The result can be hard to find even if one reads German. It is stated as (1) on page 137 of the article and mainly an application of methods developed by Flores [20, 62. Kolloqium].2 The theorem is generally known as the Hanani-Tutte theorem, though Levow [17] calls it the "van Kampen-Shapiro-Wu characterization of planar graphs" emphasizing the parallel history of the theorem in algebraic topology (ignoring Flores, however). In a recent paper [34] we introduced the name "strong Hanani-Tutte theorem" to distinguish it from a weaker version that is also often called the Hanani-Tutte theorem in the literature.

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Section: Algebraic Characterizations Of Planaritymentioning

confidence: 99%

Section: Theorem 119 (Tutte) A(g) Is a Coset Ofmentioning

confidence: 99%

Hanani-Tutte and Related Results

Schaefer

2013

Bolyai Society Mathematical Studies

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“…Theorem 2.2.a. Every n-manifold embeds into R 2n [Ka32,Wh44]. Theorem 2.2.a (as well as Theorem 2.2.b below) is proved using general position and the Whitney trick; the proof in the smooth and PL case is sketched in §4 and in [RS72, RS99, §8], respectively.…”

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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“…It is known that the number 2n + 1 in this theorem can not be reduced, a famous counterexample was given by Van Kampen [15] and Flores [4] who showed that the n-skeleton of a (2n + 2)-simplex does not embed into R 2n . But there certainly exist n-dimensional compact metric spaces 472 N. Mramor-Kosta and E. Trenklerova [2] which are embeddable into R 2n , for example, this is true for every n-dimensional manifold [14]. On the contrary, for basic embeddings Sternfeld [12] proved the following theorem.…”

Section: Introductionmentioning

confidence: 99%

On basic embeddings of compacta into the plane

Mramor-Kosta

Trenklerová

2003

Bull. Austral. Math. Soc.

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A compactum K ⊂ ℝ2 is said to be basically embedded in ℝ2 if for each continuous function f: K → ℝ there exist continuous functions g, h: ℝ → ℝ such that f(x, y) = g(x) + h(y) for each point (x, y) ∈ K. Sternfeld gave a topological characterization of compacta K which are basically embedded in ℝ2 which can be formulated in terms of special sequences of points called arrays, using arguments from functional analysis. In this paper we give a simple topological proof of the implication: if there exists an array in K of length n for any n ∈ ℕ, then K is not basically embedded.

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The Self-Intersections of a Smooth n-Manifold in 2n-Space

Cited by 112 publications

References 0 publications

Hanani-Tutte and Related Results

Hanani-Tutte and Related Results

Embedding and knotting of manifolds in Euclidean spaces

On basic embeddings of compacta into the plane

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