M. Skopenkov

A formula for the group of links in the 2-metastable
dimension.

http://arXiv.org/abs/math.GT/0610320

We present a short proof of an explicit formula for the group of
links (and also link maps) in the $2$-metastable dimension. This
improves a result of Haefliger from 1966.

Theorem. Assume that $p\le q\le m-3$ and $2p+2q\le
3m-7$. Denote by $L^m_{p,q}$ (resp. $K^m_p$) the group of smooth
embeddings $S^p\sqcup S^q\to S^m$ (resp. $S^p\to S^m$) up to
smooth isotopy. Then
$$
L^m_{p,q}\cong
\pi_p(S^{m-q-1})\oplus\pi_{p+q+2-m}(SO/SO_{m-p-1})\oplus
K^m_p\oplus K^m_q.
$$

Our approach is based on an exact sequence involving the group of
links, link maps and their relative versions. The latter are
classified via the $\beta$-invariant of Koschorke, Habegger and
Kaiser.