The Product of Nonplanar Complexes does not Imbed in 4-Space

Ummel, Brian R.

doi:10.2307/1997741

Cited by 7 publications

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“…• the k-th power of a non-planar graph (conjectured by Menger in 1929, proved in [Um78,Sk03]). The assumption of this theorem is fulfilled when d = 2k ≥ 6.…”

Section: Algorithmic Recognition Of Realizablity Of Hypergraphsmentioning

confidence: 99%

Invariants of Graph Drawings in the Plane

Skopenkov

2020

Arnold Math J.

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We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and combinatorics. We define a Z 2 -valued self-intersection invariant (i.e. the van Kampen number) and its generalizations. We present elementary formulations and arguments, so we do not require any knowledge of algebraic topology. This survey is accessible to mathematicians not specialized in any of the areas discussed. It may serve as an introduction into algebraic topology motivated by algorithmic, combinatorial and geometric problems.

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“…• the k-th power of a non-planar graph (conjectured by Menger in 1929, proved in [Um78,Sk03]). The assumption of this theorem is fulfilled when d = 2k ≥ 6.…”

Section: Algorithmic Recognition Of Realizablity Of Hypergraphsmentioning

confidence: 99%

Invariants of Graph Drawings in the Plane

Skopenkov

2020

Arnold Math J.

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“…The Product Theorem 3.2 is reduced to Proposition 2.3.b in §3.2. Theorem 3.3 (Square; [Um78,Sk03]). Any (5, 5)-product in 4-space has two triangles which have disjoint vertices but intersect.…”

Section: Realizability Of Products and The Menger Conjecturementioning

confidence: 99%

“…Proof of the Menger conjecture (see §1.4) in [Um78] works for the topological version but is complicated (one computes an obstruction via spectral sequences). Proof in [Sk03] is much simpler but for the topological version uses the Bryant approximation theorem which is not easy.…”

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confidence: 99%

Realizability of hypergraphs and Ramsey link theory

Skopenkov¹

2014

Preprint

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We present short simple proofs of Conway-Gordon-Sachs' theorem on graphs in 3-dimensional space, as well as van Kampen-Flores' and Ummel's theorems on nonrealizability of certain hypergraphs (or simplicial complexes) in 4-dimensional space. The proofs use a reduction to lower dimensions which allows to exhibit relation between these results.We present a simplified exposition accessible to non-specialists in the area and to students who know basic geometry of 3-dimensional space and who are ready to learn straightforward 4-dimensional generalizations. We use elementary language (e.g. collections of points) which allows to present the main ideas without technicalities (e.g. without using the formal definition of a hypergraph). 'It's too difficult.' 'Write simply.' 'That's hardest of all.' I. Murdoch, The Message to the Planet.

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“…Since the van Kampen obstruction is complete for d = 1, Theorem 5 is a consequence of the algebraic Menger conjecture. The Menger conjecture was first proved by Ummel [28] for product of two graphs using advanced algebraic topology techniques. M. Skopenkov [24] gave an elegant geometric proof of the more general case of product of multiple graphs.…”

Section: Introductionmentioning

confidence: 99%

On the Smith classes, the van Kampen obstruction and embeddability of $[3]*K$

Parsa

2020

Preprint

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In this survey-research paper, we first introduce the theory of Smith classes of complexes with fixed-point free, periodic maps on them. These classes, when defined for the deleted product of a simplicial complex K, are the same as the embedding classes of K. Embedding classes, in turn, are generalizations of the van Kampen obstruction class for embeddability of a d-dimensional complex K into the Euclidean 2d-space. All of these concepts will be introduced in simple terms.Second, we use the theory introduced in the first part to relate the embedding classes (or the special Smith classes) of the the complex [3] * K with the embedding classes of K. Here [3] * K is the join of K with a set of three points.Specifically, we prove that if the m-th embedding class of K is nonzero, then the (m+2)-nd embedding class of [3] * K is non-zero. We also prove some of the consequences of this theorem for the embeddability of [3] * K.

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The Product of Nonplanar Complexes does not Imbed in 4-Space

Abstract: We prove that if Kx and K2 are nonplanar simplicial complexes, then Kx X K2 does not imbed in R4.

Cited by 7 publications

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Invariants of Graph Drawings in the Plane

Invariants of Graph Drawings in the Plane

Realizability of hypergraphs and Ramsey link theory

On the Smith classes, the van Kampen obstruction and embeddability of $[3]*K$

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