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Suspension theorems for links and link maps

Author: Mikhail Skopenkov
Journal: Proc. Amer. Math. Soc. 137 (2009), 359-369
MSC (2000): Primary 57Q45, 57R40; Secondary 55P40, 57Q30
Published electronically: August 26, 2008
MathSciNet review: 2439461
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Abstract: We present a new short proof of the explicit formula for the group of links (and also link maps) in the ``quadruple point free'' dimension. Denote by $ L^m_{p,q}$ (respectively, $ C^{m-p}_p$) the group of smooth embeddings $ S^p\sqcup S^q\to S^m$ (respectively, $ S^p\to S^m$) up to smooth isotopy. Denote by $ LM^m_{p,q}$ the group of link maps $ S^p\sqcup S^q\to S^m$ up to link homotopy.

Theorem 1. If $ p\le q\le m-3$ and $ 2p+2q\le 3m-6$, then

$\displaystyle L^m_{p,q}\cong \pi_p(S^{m-q-1})\oplus\pi_{p+q+2-m}(SO/SO_{m-p-1})\oplus C^{m-p}_p\oplus C^{m-q}_q. $

Theorem 2. If $ p, q\le m-3$ and $ 2p+2q\le 3m-5$, then $ LM^m_{p,q}\cong \pi^S_{p+q+1-m}$.

Our approach is based on the use of the suspension operation for links and link maps, and suspension theorems for them.

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Additional Information

Mikhail Skopenkov
Affiliation: Department of Differential Geometry, Faculty of Mechanics and Mathematics, Moscow State University, 119992, Moscow, Russia

Keywords: Link, link map, link homotopy, homotopy groups, Stiefel manifold, suspension, the EHP sequence, engulfing, linking number, alpha-invariant, beta-invariant
Received by editor(s): May 15, 2006
Received by editor(s) in revised form: November 1, 2007
Published electronically: August 26, 2008
Additional Notes: The author was supported in part by INTAS grant 06-1000014-6277, Russian Foundation of Basic Research grants 05-01-00993-a, 06-01-72551-NCNIL-a, 07-01-00648-a, President of the Russian Federation grant NSh-4578.2006.1, Agency for Education and Science grant RNP-, and Moebius Contest Foundation for Young Scientists.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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