  We synthesize work of U. Koschorke on link maps and work of B. Johnson on the
derivatives of the identity functor in homotopy theory. The result can be
viewed in two ways: (1) As a generalization of Koschorke's "higher Hopf
invariants", which themselves can be viewed as a generalization of Milnor's
invariants of link maps in Euclidean space; and (2) As a stable range
description, in terms of bordism, of the cross effects of the identity functor
in homotopy theory evaluated at spheres. We also show how our generalized
Milnor invariants fit into the framework of a multivariable manifold calculus
of functors, as developed by the author and Voli\'{c}, which is itself a
generalization of the single variable version due to Weiss and Goodwillie.
