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 $L^m_{p,q}$ (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.