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

This article is cited in 22 scientific papers (total in 24 papers)

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

Full text: PDF file (678 kB)
References: PDF file   HTML file

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
Received: 12.08.1999

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

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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet
    2. Skopenkov, A, “On the generalized Massey-Rolfsen invariant for link maps”, Fundamenta Mathematicae, 165:1 (2000), 1  mathscinet  zmath  isi  elib
    3. Repovs, D, “On projected embeddings and desuspending the alpha-invariant”, Topology and Its Applications, 124:1 (2002), 69  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    4. Skopenkov, A, “On the Haefliger-Hirsch-Wu invariants for embeddings and immersions”, Commentarii Mathematici Helvetici, 77:1 (2002), 78  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    5. Skopenkov M., “Embedding products of graphs into Euclidean spaces”, Fund. Math., 179:3 (2003), 191–198  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    6. Skopenkov M., “On approximability by embeddings of cycles in the plane”, Topology Appl., 134:1 (2003), 1–22  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    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  mathscinet  zmath  isi  elib
    8. I. Maleshich, P. E. Pushkar', D. Repovš, “On Eversion of Spheres”, Proc. Steklov Inst. Math., 247 (2004), 135–142  mathnet  mathscinet  zmath
    9. M. Cencelj, D. Repovš, A. B. Skopenkov, “On the Browder–Levine–Novikov Embedding Theorems”, Proc. Steklov Inst. Math., 247 (2004), 259–268  mathnet  mathscinet  zmath
    10. S. A. Melikhov, “Isotopic and continuous realizability of maps in the metastable range”, Sb. Math., 195:7 (2004), 983–1016  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    11. S. A. Melikhov, “On isotopic realizability of maps factored through a hyperplane”, Sb. Math., 195:8 (2004), 1117–1163  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    12. A. B. Skopenkov, “Vokrug kriteriya Kuratovskogo planarnosti grafov”, Matem. prosv., ser. 3, 9, Izd-vo MTsNMO, M., 2005, 116–128  mathnet
    13. A. Yu. Volovikov, E. V. Shchepin, “Antipodes and embeddings”, Sb. Math., 196:1 (2005), 1–28  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    14. Cencelj, M, “On embeddings of tori in Euclidean spaces”, Acta Mathematica Sinica-English Series, 21:2 (2005), 435  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    15. Goncalves, D, “Embeddings of homology equivalent manifolds with boundary”, Topology and Its Applications, 153:12 (2006), 2026  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    16. Repovs, D, “On basic embeddings into the plane”, Rocky Mountain Journal of Mathematics, 36:5 (2006), 1665  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    17. Skopenkov, A, “A new invariant and parametric connected sum of embeddings”, Fundamenta Mathematicae, 197 (2007), 253  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    18. Eleftheriou, PE, “A semi-linear group which is not affine”, Annals of Pure and Applied Logic, 156:2–3 (2008), 287  crossref  mathscinet  zmath  isi  scopus  scopus
    19. Skopenkov, A, “A classification of smooth embeddings of 3-manifolds in 6-space”, Mathematische Zeitschrift, 260:3 (2008), 647  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    20. Skopenkov, M, “Suspension theorems for links and link maps”, Proceedings of the American Mathematical Society, 137:1 (2009), 359  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    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  crossref  mathscinet  zmath  isi
    22. Skopenkov A., “A classification of smooth embeddings of 4-manifolds in 7-space, I”, Topology and Its Applications, 157:13 (2010), 2094–2110  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    24. D. Repovš, M. B. Skopenkov, M. Cencelj, “Classification of knotted tori in 2-metastable dimension”, Sb. Math., 203:11 (2012), 1654–1681  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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