Template:Avramidi&Nguyen Phan2021

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G. Avramidi, T. Tam Nguyen Phan, Fungible obstructions to embedding 2-complexes, arxiv preprint.

[edit] 1 Comment

The idea of relating non-embeddability to linkless embeddings used in \S2.2, \S5.2 is introduced in [Skopenkov2003, Example 2]. This idea is applied to

  • the 2-skeleton of 6-simplex, as in \S2.2, in [Skopenkov2014a, \S2.3].
  • the product of graphs (proof of the generalized Menger conjecture) in [Skopenkov2003, Theorem 1].

A proof and a generalization of the result mentioned in \S2.3, On Figure 3(c), is presented in [Karasev&Skopenkov2020].

NP hardness results based on the Freedman-Kruskal-Teichner example and its generalizations are presented in [Matousek&Tancer&Wagner2008], [Skopenkov&Tancer2017].

A simpler proof and an improvement of the Freedman-Kruskal-Teichner example (a 2-polyhedron not almost embeddable in \R^4 but for which the van Kampen obstruction is zero) are given in [Avvakumov&Mabillard&Wagner&Skopenkov2015, Theorem 1.6 and \S2.2].

Example in [Segal&Skopenkov&Spiez1998, p. 338] improves the Freedman-Kruskal-Teichner example in a different direction (a 2-polyhedron almost embeddable but not embeddable in \R^4). For this improvement the p-fold deleted product obstructions vanish as explained in [Skopenkov2006, \S5, The Generalized Haefliger-Wu invariant].

[edit] 2 References

  • [Avvakumov&Mabillard&Wagner&Skopenkov2015] S. Avvakumov, I. Mabillard, A. Skopenkov and U. Wagner. Eliminating Higher-Multiplicity Intersections, III. Codimension 2, Israel J. Math. (2021). arxiv preprint.
  • [Karasev&Skopenkov2020] R. Karasev and A. Skopenkov. Some `converses' to intrinsic linking theorems. Discr. Comp. Geom., 70:3 (2023), 921--930. arxiv preprint.
  • [Matousek&Tancer&Wagner2008] J. Matousek, M. Tancer, U. Wagner. Hardness of embedding simplicial complexes in \R^d, J. Eur. Math. Soc. 13:2 (2011), 259--295. arxiv preprint.
  • [Parsa2015] S. Parsa, On links of vertices in simplicial d-complexes embeddable in the euclidean 2d-space, Discrete Comput. Geom. 59:3 (2018), 663--679. arxiv preprint.
  • [Segal&Skopenkov&Spiez1998] J. Segal, A. Skopenkov and S. Spie\. z, Embeddings of polyhedra in \R^m and the deleted product obstruction, Topol. Appl. 85 (1998), 225-234.
  • [Skopenkov&Tancer2017] A. Skopenkov and M. Tancer. Hardness of almost embedding simplicial complexes in \R^d, Discr. Comp. Geom., 61:2 (2019), 452--463. arxiv preprint.
  • [Skopenkov2003] M. Skopenkov, Embedding products of graphs into Euclidean spaces, Fund. Math. 179 (2003) 191-198. arxiv preprint.
  • [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
  • [Skopenkov2014a] A. Skopenkov, Realizability of hypergraphs and Ramsey link theory. arxiv preprint.
  • [Skopenkov2018] A. Skopenkov. A short exposition of S. Parsa's theorems on intrinsic linking and non-realizability. Discr. Comp. Geom. 65:2 (2021), 584-585. Full version: arxiv preprint.
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