## Suspension theorems for links and link maps

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- by Mikhail Skopenkov
- Proc. Amer. Math. Soc.
**137**(2009), 359-369 - DOI: https://doi.org/10.1090/S0002-9939-08-09455-0
- Published electronically: August 26, 2008
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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* \begin{equation*} 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. \end{equation*}

**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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## Bibliographic Information

**Mikhail Skopenkov**- Affiliation: Department of Differential Geometry, Faculty of Mechanics and Mathematics, Moscow State University, 119992, Moscow, Russia
- Email: skopenkov@rambler.ru
- 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-2.1.1.7988, and Moebius Contest Foundation for Young Scientists.
- Communicated by: Alexander N. Dranishnikov
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 359-369 - MSC (2000): Primary 57Q45, 57R40; Secondary 55P40, 57Q30
- DOI: https://doi.org/10.1090/S0002-9939-08-09455-0
- MathSciNet review: 2439461