Homology groups of types in stable theories and the Hurewicz correspondence

TL;DR

This paper explicitly describes the homomorphism groups of strong types in stable theories, establishing an isomorphism with automorphism groups of algebraic closures, under certain triviality assumptions, and introduces the Hurewicz correspondence analogy.

Contribution

It provides a new explicit description of homology groups of types in stable theories and establishes the Hurewicz correspondence relating these groups to automorphism groups.

Findings

H_n(p) is isomorphic to automorphism groups of algebraic closures.

H_n(p) groups are abelian.

The description applies under specific triviality conditions for H_i(q).

Abstract

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.