  We give an explicit description of the homomorphism group H_n(p) of a strong
type p in any stable theory under the assumption that for every non-forking
extension q of p the groups H_i(q) are trivial for i at least 2 but less than
n. The group H_n(p) turns out to be isomorphic to the automorphism group of a
certain piece of the algebraic closure of n independent realizations of p; it
was shown earlier by the authors that such a group must be abelian. We call
this the "Hurewicz correspondence" in analogy with the Hurewicz Theorem in
algebraic topology.
