James construction, Fox torus homotopy groups, and Hopf invariants

For any pointed space Y, the James construction J(Y) is a free monoid generated by Y and its suspension Sigma J(Y) has the homotopy type Sigma V-n >= 1 Lambda Y-n, where Lambda Y-n denotes the n-fold smash product of Y. For any path connected space X, the homotopy groups {pi(i)(X)} can be encoded, as sets, in the homotopy group [J(S-1), Omega X]. In this paper, we make use of the Fox torus homotopy groups tau(n)(X) and their generalizations to encode the homotopy groups of X by embedding [J(n)(S-1), Omega X] into tau(n+1)(X) as a subgroup, where J(n)(Y) is the n-th stage of the James filtration of J(Y). We generalize this embedding for generalized Fox groups tau(X) and describe the image of [J(n)(Y), Omega X] in tau(n)(Y)(X). Finally, we describe the generalized Hopf invariants in the context of generalized torus homotopy groups in connection with the James construction.