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 Uspekhi Mat. Nauk, 1999, Volume 54, Issue 6(330), Pages 61–108 (Mi umn230)

New results on embeddings of polyhedra and manifolds in Euclidean spaces

D. Repovša, A. B. Skopenkovb

a University of Ljubljana
b Advanced Educational Scientific Center of M. V. Lomonosov Moscow State University — A. N. Kolmogorov School

Abstract: The aim of this survey is to present several classical results on embeddings and isotopies of polyhedra and manifolds in $\mathbb R^m$. We also describe the revival of interest in this beautiful branch of topology and give an account of new results, including an improvement of the Haefliger–Weber theorem on the completeness of the deleted product obstruction to embeddability and isotopy of highly connected manifolds in $\mathbb R^m$ (Skopenkov) as well as the unimprovability of this theorem for polyhedra (Freedman, Krushkal, Teichner, Segal, Skopenkov, and Spiez) and for manifolds without the necessary connectedness assumption (Skopenkov). We show how algebraic obstructions (in terms of cohomology, characteristic classes, and equivariant maps) arise from geometric problems of embeddability in Euclidean spaces. Several classical and modern results on completeness or incompleteness of these obstructions are stated and proved. By these proofs we illustrate classical and modern tools of geometric topology (engulfing, the Whitney trick, van Kampen and Casson finger moves, and their generalizations).

DOI: https://doi.org/10.4213/rm230

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English version:
Russian Mathematical Surveys, 1999, 54:6, 1149–1196

Bibliographic databases:

UDC: 515.14+515.16
MSC: Primary 57Q35, 57R40; Secondary 57R42, 57R52, 55S35, 57Q30, 57R20, 57N35, 52B11

Citation: D. Repovš, A. B. Skopenkov, “New results on embeddings of polyhedra and manifolds in Euclidean spaces”, Uspekhi Mat. Nauk, 54:6(330) (1999), 61–108; Russian Math. Surveys, 54:6 (1999), 1149–1196

Citation in format AMSBIB
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This publication is cited in the following articles:
1. D. Repovsh, A. B. Skopenkov, “Teoriya prepyatstvii dlya nachinayuschikh”, Matem. prosv., ser. 3, 4, MTsNMO, M., 2000, 154–180
2. Skopenkov, A, “On the generalized Massey-Rolfsen invariant for link maps”, Fundamenta Mathematicae, 165:1 (2000), 1
3. Repovs, D, “On projected embeddings and desuspending the alpha-invariant”, Topology and Its Applications, 124:1 (2002), 69
4. Skopenkov, A, “On the Haefliger-Hirsch-Wu invariants for embeddings and immersions”, Commentarii Mathematici Helvetici, 77:1 (2002), 78
5. Skopenkov M., “Embedding products of graphs into Euclidean spaces”, Fund. Math., 179:3 (2003), 191–198
6. Skopenkov M., “On approximability by embeddings of cycles in the plane”, Topology Appl., 134:1 (2003), 1–22
7. Malešič J., Repovš D., Skopenkov A., “On incompleteness of the deleted product obstruction for embeddability”, Bol. Soc. Mat. Mexicana (3), 9:1 (2003), 165–170
8. I. Maleshich, P. E. Pushkar', D. Repovš, “On Eversion of Spheres”, Proc. Steklov Inst. Math., 247 (2004), 135–142
9. M. Cencelj, D. Repovš, A. B. Skopenkov, “On the Browder–Levine–Novikov Embedding Theorems”, Proc. Steklov Inst. Math., 247 (2004), 259–268
10. S. A. Melikhov, “Isotopic and continuous realizability of maps in the metastable range”, Sb. Math., 195:7 (2004), 983–1016
11. S. A. Melikhov, “On isotopic realizability of maps factored through a hyperplane”, Sb. Math., 195:8 (2004), 1117–1163
12. A. B. Skopenkov, “Vokrug kriteriya Kuratovskogo planarnosti grafov”, Matem. prosv., ser. 3, 9, Izd-vo MTsNMO, M., 2005, 116–128
13. A. Yu. Volovikov, E. V. Shchepin, “Antipodes and embeddings”, Sb. Math., 196:1 (2005), 1–28
14. Cencelj, M, “On embeddings of tori in Euclidean spaces”, Acta Mathematica Sinica-English Series, 21:2 (2005), 435
15. Goncalves, D, “Embeddings of homology equivalent manifolds with boundary”, Topology and Its Applications, 153:12 (2006), 2026
16. Repovs, D, “On basic embeddings into the plane”, Rocky Mountain Journal of Mathematics, 36:5 (2006), 1665
17. Skopenkov, A, “A new invariant and parametric connected sum of embeddings”, Fundamenta Mathematicae, 197 (2007), 253
18. Eleftheriou, PE, “A semi-linear group which is not affine”, Annals of Pure and Applied Logic, 156:2–3 (2008), 287
19. Skopenkov, A, “A classification of smooth embeddings of 3-manifolds in 6-space”, Mathematische Zeitschrift, 260:3 (2008), 647
20. Skopenkov, M, “Suspension theorems for links and link maps”, Proceedings of the American Mathematical Society, 137:1 (2009), 359
21. Skopenkov A., “Embeddings of $k$-Connected $n$-Manifolds into $\Bbb R^{2n-k-1}$”, Proceedings of the American Mathematical Society, 138:9 (2010), 3377–3389
22. Skopenkov A., “A classification of smooth embeddings of 4-manifolds in 7-space, I”, Topology and Its Applications, 157:13 (2010), 2094–2110
23. Crowley D., Skopenkov A., “A Classification of Smooth Embeddings of Four-Manifolds in Seven-Space, II”, Internat J Math, 22:6 (2011), 731–757
24. D. Repovš, M. B. Skopenkov, M. Cencelj, “Classification of knotted tori in 2-metastable dimension”, Sb. Math., 203:11 (2012), 1654–1681
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