The wikipedia article on the Jhomomorphism says that "the cokernel of the Jhomomorphism is of interest for counting exotic spheres". I'd like to think this makes some sort of philosophical sense; as I understand it, the homomorphism comes from a Hopf construction, which isn't really a smooth sort of thing, but on the other hand it still feels like constructions using the (stable) (special) orthogonal group should somehow keep us in the nonexotic world. Is this at all close to the right intuition?
