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Homology groups of types in stable theories and the Hurewicz correspondence
Goodrick, J.,Kim, B.,Kolesnikov, A. North-Holland ; Elsevier Science Ltd 2017 Annals of pure and applied logic Vol.168 No.9
<P>We give an explicit description of the homology 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 2 <= i < n. The group H-n(p) turns out to be isomorphic to the automorphism group of a certain part of the algebraic closure of n independent realizations of p; it follows from the authors' earlier work that such a group must be abelian. We call this the 'Hurewicz correspondence' by analogy with the Hurewicz Theorem in algebraic topology. (c) 2017 Elsevier B.V. All rights reserved.</P>