  We introduce several homotopy equivalence relations for proper holomorphic
mappings between balls. We provide examples showing that the degree of a
rational proper mapping between balls (in positive codimension) is not a
homotopy invariant. In domain dimension at least 2, we prove that the set of
homotopy classes of rational proper mappings from a ball to a higher
dimensional ball is finite. By contrast, when the target dimension is at least
twice the domain dimension, it is well known that there are uncountably many
spherical equivalence classes. We generalize this result by proving that an
arbitrary homotopy of rational maps whose endpoints are spherically
inequivalent must contain uncountably many spherically inequivalent maps. We
introduce Whitney sequences, a precise analogue (in higher dimensions) of the
notion of finite Blaschke product (in one dimension). We show that terms in a
Whitney sequence are homotopic to monomial mappings, and we establish an
additional result about the target dimensions of such homotopies.
