Lascar groups and the first homology groups of strong types in rosy theories

Abstract

For a rosy theory, we give a canonical surjective homomorphism from a Lascar group over $A=\acl^{eq}(A)$ to a first homology group of a strong type over $A$, and we describe its kernel by an invariant equivalence relation. As a consequence, we show that the first homology groups of strong types in rosy theories have the cardinalities of one or at least $2^{\aleph_0}$. We give two examples of rosy theories having non trivial first homology groups of strong types over $\acl^{eq}(\emptyset)$. In these examples, these two homology groups are exactly isomorphic to their Lascar group over $\acl^{eq}(\emptyset)$.