Hardness of almost embedding simplicial complexes in $\mathbb R^d$

Skopenkov, Arkadiy; Tancer, Martin

doi:10.48550/arxiv.1703.06305

Cited by 3 publications

(4 citation statements)

References 16 publications

(34 reference statements)

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“…M. Tancer suggests that it is plausible to approach the conjecture the same way as in [MTW11,ST17]. Namely, one can possibly triangulate the gadgets in advance and glue them together so that the 'embeddable gadgets' would be linearly embeddable with respect to the prescribed triangulations.…”

Section: Algorithmic Recognition Of Almost Realizablity Of Hypergraphsmentioning

confidence: 99%

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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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Section: Algorithmic Recognition Of Almost Realizablity Of Hypergraphsmentioning

confidence: 99%

“…Its generalizations (to Z r -valued invariants and to cohomological obstructions) are defined and used to obtain elementary formulations and proofs of §1 and §2 mentioned above. For applications of another generalizations see [Sk16,§4], [Sk16',ST17]. For invariants of plane curves and caustics see [Ar95] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Invariants of Graph Drawings in the Plane

Skopenkov

2020

Arnold Math J.

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“…Lemma 1.3 is a stronger version of [SS92, Lemma 1.4] (in [SS92] it was assumed that f | Σ ℓ is an embedding). This stronger version is essentially proved in [ST17,§3], see a sketch of a proof in §2. (The proof of the stronger version uses a simpler argument: application of Lemma 2.4 instead of the cohomological Smith index as in [SS92,§1].…”

Section: Introduction and Main Resultsmentioning

confidence: 92%

Some `converses' to intrinsic linking theorems

Карасев¹,

Skopenkov²

2020

Preprint

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A low-dimensional version of our main result is the following 'converse' of the Conway-Gordon-Sachs Theorem on intrinsic linking of the graph K 6 in 3-space:For any integer z there are 6 points 1, 2, 3, 4, 5, 6 in 3-space, of which every two i, j are joint by a polygonal line ij, the interior of one polygonal line is disjoint with any other polygonal line, the linking coefficient of any pair disjoint 3-cycles except for {123, 456} is zero, and for the exceptional pair {123, 456} is 2z + 1.We prove a higher-dimensional analogue, which is a 'converse' of a lemma by Segal-Spież. * We would like to thank F. Frick for helpful discussions.

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“…(a) Our proof of the Singular Borromean Rings Lemma 2.4 for n = 2 and l = 1 gives a shorter, elementary proof of the result from[FKT] mentioned before Theorem 1.6. (b) In[ST17], the Singular Borromean Rings Lemma 2.4 is used to study algorithmic aspects of almost 2-embeddability of complexes in R d . (c) The analogue of the Singular Borromean Rings Lemma 2.4 for n = l + 1 = 1 is true, although our proof does not work for this case.…”

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confidence: 99%

Eliminating higher-multiplicity intersections. III. Codimension 2

et al. 2021

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We study conditions under which a finite simplicial complex K can be mapped to R d without higher-multiplicity intersections. An almost r-embedding is a map f : K → R d such that the images of any r pairwise disjoint simplices of K do not have a common point. We show that if r is not a prime power and d ≥ 2r + 1, then there is a counterexample to the topological Tverberg conjecture, i.e., there is an almost r-embedding of the (d + 1)(r − 1)-simplex in R d . This improves on previous constructions of counterexamples (for d ≥ 3r) based on a series of papers by M. Özaydin, M. Gromov, P. Blagojević, F. Frick, G. Ziegler, and the second and fourth present authors.The counterexamples are obtained by proving the following algebraic criterion in codimension 2: If r ≥ 3 and if K is a finite 2(r − 1)-complex then there exists an almost rembedding K → R 2r if and only if there exists a general position PL map f : K → R 2r such that the algebraic intersection number of the f -images of any r pairwise disjoint simplices of K is zero. This result can be restated in terms of cohomological obstructions or equivariant maps, and extends an analogous codimension 3 criterion by the second and fourth authors. As an application we classify ornaments f : S 3 S 3 S 3 → R 5 up to ornament concordance.It follows from work of M. Freedman, V. Krushkal, and P. Teichner that the analogous criterion for r = 2 is false. We prove a lemma on singular higher-dimensional Borromean rings, yielding an elementary proof of the counterexample.

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Hardness of almost embedding simplicial complexes in $\mathbb R^d$

Cited by 3 publications

References 16 publications

Invariants of Graph Drawings in the Plane

Invariants of Graph Drawings in the Plane

Some `converses' to intrinsic linking theorems

Eliminating higher-multiplicity intersections. III. Codimension 2

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