Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Embeddings of $ k$-connected $ n$-manifolds into $ \mathbb{R}^{2n-k-1}$

Author: A. Skopenkov
Journal: Proc. Amer. Math. Soc. 138 (2010), 3377-3389
MSC (2010): Primary 57R40, 57Q37; Secondary 57R52
Published electronically: May 4, 2010
MathSciNet review: 2653966
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain estimations for isotopy classes of embeddings of closed $ k$-connected $ n$-manifolds into $ \mathbb{R}^{2n-k-1}$ for $ n\ge 2k+6$ and $ k\ge 0$. This is done in terms of an exact sequence involving the Whitney invariants and an explicitly constructed action of $ H_{k+1}(N;\mathbb{Z}_{2})$ on the set of embeddings. The proof involves a reduction to the classification of embeddings of a punctured manifold and uses the parametric connected sum of embeddings.

Corollary. Suppose that $ N$ is a closed almost parallelizable $ k$-connected $ n$-manifold and $ n\ge 2k+6\ge 8$. Then the set of isotopy classes of embeddings $ N\to \mathbb{R}^{2n-k-1}$ is in 1-1 correspondence with $ H_{k+2}(N;\mathbb{Z} _{2})$ for $ n-k=4s+1$.

References [Enhancements On Off] (What's this?)

  • [BG71] J. C. Becker and H. H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. of the Amer. Math. Soc. 27:2 (1971), 405-410. MR 0268903 (42:3800)
  • [BH70] J. Boéchat and A. Haefliger, Plongements différentiables de variétés orientées de dimension $ 4$ dans $ \mathbb{R} ^{7}$, Essays on topology and related topics, Springer, 1970, 156-166. MR 0270384 (42:5273)
  • [CS08] D. Crowley and A. Skopenkov, A classification of smooth embeddings of $ 4$-manifolds in $ 7$-space, II, submitted, arXiv:math/0808.1795.
  • [HCEC] High_codimension_embeddings:_classification
  • [Hu69] J. F. P. Hudson, Piecewise-Linear Topology, Benjamin, New York, Amsterdam, 1969. MR 0248844 (40:2094)
  • [PCS]
  • [Po85] M. M. Postnikov, Homotopy theory of CW-complexes, Nauka, Moscow, 1985 (in Russian).
  • [Pr07] V. V. Prasolov, Elements of Homology Theory, Graduate Studies in Mathematics, Amer. Math. Soc., Providence, RI, 2007. Earlier Russian version available at MR 2313004 (2008d:55001)
  • [Ri70] R. D. Rigdon, Thesis, Ph.D. Thesis (1970).
  • [RS99] D. Repovš and A. Skopenkov, New results on embeddings of polyhedra and manifolds into Euclidean spaces, Uspekhi Mat. Nauk 54:6 (1999), 61-109 (in Russian); English transl., Russ. Math. Surv. 54:6 (1999), 1149-1196. MR 1744658 (2001c:57028)
  • [Sa99] O. Saeki, On punctured $ 3$-manifolds in $ 5$-sphere, Hiroshima Math. J. 29 (1999), 255-272. MR 1704247 (2000h:57045)
  • [Sk05] A. Skopenkov, A classification of smooth embeddings of $ 4$-manifolds in $ 7$-space, submitted, arXiv:math/0512594.
  • [Sk06] A. Skopenkov, Classification of embeddings below the metastable dimension, submitted, arXiv:math/0607422.
  • [Sk07] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253-269. arXiv:math/0509621. MR 2365891 (2008k:57044)
  • [Sk08] 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, Cambridge Univ. Press, Cambridge, 2008, 248-342. arXiv:math/0604045. MR 2388495 (2009e:57040)
  • [Sk081] A. Skopenkov, A classification of smooth embeddings of $ 3$-manifolds in $ 6$-space, Math. Zeitschrift 260:3 (2008), 647-672. arXiv:math/0603429. MR 2434474
  • [Sk] A. Skopenkov, Embeddings of $ k$-connected $ n$-manifolds into $ R^{2n-k-1}$; arXiv:math/ 0812.0263.
  • [Vr89] J. Vrabec, Deforming a PL submanifold of Euclidean space into a hyperplane, Trans. Amer. Math. Soc. 312:1 (1989), 155-178. MR 937253 (89g:57029)
  • [Wh50] J.H.C. Whitehead, A certain exact sequence, Annals of Math. (2) 52 (1950), 51-110. MR 0035997 (12:43c)
  • [Ya83] T. Yasui, On the map defined by regarding embeddings as immersions, Hiroshima Math. J. 13 (1983), 457-476. MR 725959 (85g:57016)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 57R40, 57Q37, 57R52

Retrieve articles in all journals with MSC (2010): 57R40, 57Q37, 57R52

Additional Information

A. Skopenkov
Affiliation: Department of Differential Geometry, Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia 119992 – and – Independent University of Moscow, B. Vlasyevskiy, 11, 119002, Moscow, Russia

Keywords: Embedding, self-intersection, isotopy, parametric connected sum
Received by editor(s): December 16, 2008
Received by editor(s) in revised form: December 31, 2009
Published electronically: May 4, 2010
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society