J-homomorphism

Summary

The J-homomorphism $J \colon \pi_r(SO(n)) \to \pi_{r+n}(S^n)$ turns a family of rotations of $\R^n$ parametrised by an $r$-sphere into a map of spheres. It connects the homotopy of the rotation groups, known through Bott periodicity, with the homotopy groups of spheres. In the stable range it gives a homomorphism $\pi_r(SO) \to \pi_r^s$ into the stable homotopy groups of spheres, whose image is the image of J.