Homology groups of types in model theory and the computation of H2(p)

We present definitions of homology groups Hn(p), n ≥ 0, associated to a complete type p. We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group H2(p) for strong types in stable theories and show that any profinite abelian group can occur as the group H2(p). The work described in this paper was originally inspired by Hrushovski’s discovery [7] of striking connections between amalgamation properties and definable groupoids in models of a stable first-order theory. Amalgamation properties have already been much studied by researchers in simple theories. The Independence Theorem, or 3-amalgamation, was used to construct canonical bases for types in such theories in [5], and in [1], Hrushovski’s group configuration theorem for stable theories was generalized to simple theories under the assumption of 4-amalgamation over sets containing models. In [10], the n-amalgamation hierarchy was studied systematically. See Section 1 below for a precise definition of n-amalgamation. In [7], Hrushovski showed that if a stable theory fails 3-uniqueness, then there must exist a groupoid whose sets of objects and morphisms, as well as the composition of morphisms, are definable in the theory. In [6], an explicit construction of such a groupoid was given and it was shown in [3] that the group of automorphisms of each object of such a groupoid must be abelian profinite. The morphisms in the groupoid construction in [6] arise as equivalence classes of “paths”, defined in a model-theoretic way. In some sense, the groupoid construction paralleled that of the construction of a fundamental groupoid in a topological space. Thus it seemed natural to ask whether it is possible to define the notion of a homology group in model-theoretic context and, if so, whether such homology groups are related to the groups described in [6, 3]. In the present paper, we define homology groups Hn(p) for a complete strong type p in any rosy theory. We show that if T has k-amalgamation for every k between 1 and n + 2, then Hn(p) = 0 for any complete type p. This leaves open the question of the converse, whether the triviality of all homology groups could imply n-amalgamation for every n. In this paper, we make a step in this