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Homology groups of types in stable theories and the Hurewicz correspondence

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.