Homology groups of types in stable theories and the Hurewicz correspondence

We give an explicit description of the homology group Hn(p) of a strong type p in any stable theory under the assumption that for every non-forking extension q of p the groups Hi(q) are trivial for 2 ≤ i < n. The group Hn(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. The present paper is a part of the project to study type amalgamation properties in first-order theories by means of homology groups of types. Roughly speaking (more precise definitions are recalled below in Section 1), a strong type p is said to have n-amalgamation if commuting systems of elementary embeddings among algebraic closures of proper subsets of the set of n independent realizations of p can always be extended to the algebraic closure of all n realizations. The type p has n-uniqueness if this extension is essentially unique. Generalized amalgamation properties for systems of models were introduced by Shelah in [12] and played an important role in [13]. The type amalgamation properties were studied extensively by Hrushovski in [9] and applications were given. In fact, the type amalgamation properties have been used in model theory at least as far back as Hrushovski’s classification of trivial totally categorical theories in [8]. In the previous paper [5], we introduced a notion of homology groups for a complete strong type in any stable, or even rosy, first-order theory. The idea was that these homology groups should measure information about the amalgamation properties of the type p. We could prove that if p has n-amalgamation for all n, then Hn(p) = 0 for every n, and that the failure of 4-amalgamation (equivalently, the failure of 3-uniqueness) The second author was supported by NRF of Korea grant 2013R1A1A2073702, and Samsung Science Technology Foundation under Project Number SSTFBA1301-03.