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Proceedings of the American Mathematical Society
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
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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$.


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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
Email: skopenko@mccme.ru

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10425-0
PII: S 0002-9939(10)10425-0
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.