Suspension theorems for links and link maps

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 $L_{p,q}^m$ (respectively, $C_p^{m-p}$) the group of smooth embeddings $S^p\bigsqcup S^q\to S^m$ (respectively, $S^p\to S^m$) up to smooth isotopy. Denote by $L{M}_{p,q}^m$ the group of link maps $S^p\bigsqcup S^q\to S^m$ 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 π_p(S^{m-q-1})\oplusπ_{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 $L{M}_{p,q}^m\cong {\pi }_{p+q+1-m}^S$. Our approach is based on the use of the suspension operation for links and link maps, and suspension theorems for them.

in English and in Russian, 12 pages, 3 figures; minor correction in the definition of the vertical homomorphisms in Theorem 3.5

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