Template:Avramidi&Nguyen Phan2021
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 2.2, 5.2 is introduced in [Skopenkov2003, Example 2]. This idea is applied to
- the 2-skeleton of 6-simplex, as in 2.2, in [Skopenkov2014a, 2.3].
- the product of graphs (proof of the generalized Menger conjecture) in [Skopenkov2003, Theorem 1].
- the join with triple-point set in [Parsa2015], [Skopenkov2018].
A proof and a generalization of the result mentioned in 2.3, 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 but for which the van Kampen obstruction is zero) are given in [Avvakumov&Mabillard&Wagner&Skopenkov2015, Theorem 1.6 and 2.2].
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 ). For this improvement the -fold deleted product obstructions vanish as explained in [Skopenkov2006, 5, 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 , J. Eur. Math. Soc. 13:2 (2011), 259--295. arxiv preprint.
- [Parsa2015] S. Parsa, On links of vertices in simplicial -complexes embeddable in the euclidean -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 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 , 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.