To each three-component link in the 3-sphere, we associate a geometrically
natural characteristic map from the 3-torus to the 2-sphere, and show that the
pairwise linking numbers and Milnor triple linking number that classify the
link up to link homotopy correspond to the Pontryagin invariants that classify
its characteristic map up to homotopy. This can be viewed as a natural
extension of the familiar fact that the linking number of a two-component link
in 3-space is the degree of its associated Gauss map from the 2-torus to the
2-sphere.