Given a topological group $G$ one forms the commutator $c\colon G\times G\rightarrow G$, $(x,y)\mapsto xyx^{-1}y^{-1}$. This map then factors through the smash $G\wedge G$. This map is the most general form of a Samelson product in the group $G$ and is of much interest. In particular, it has finite order for any compact simply connected Lie group. Bounds on this order may be obtained by studying the Samelson product of two maps $\alpha:X\rightarrow G$, $\beta:Y\rightarrow G$, defined to be the composite
$\langle \alpha,\beta\rangle \colon X\wedge Y\xrightarrow{\alpha\wedge\beta}G\wedge G\xrightarrow{c}G$
The case when $X=S^r$, $Y=S^s$ being particularly appealing.
Bott studied Samelson products in the homotopy of the unitary groups $U(n)$ and symplectic groups $Sp(n)$ and discovered some interesting relations int he metastable homotopy of these groups.
But what is know of Samelson products in the special orthogonal groups $SO(n)$?
There are certain papers by Kishimoto, Hamanaka and Kono that investigate p-local Samelson products. Similarly, Mahowald has discovered a non-trivial product in SO(2n). But what is known in general? The metastable homotopy of $SO(n)$ is particularly rich and it seems that there may be important connections to other problems such as the unstable J-homomorphism. Is anything concrete known?
