The Lascar groups and the first homology groups in model theory
Annals of Pure and Applied Logic
Attribution-Non Commercial No Derivatives 4.0 International License (https://creativecommons.org/licenses/by-nc-nd/4.0/). firstname.lastname@example.org https://eprints.whiterose.ac.uk/ Reuse This article is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) licence. This licence only allows you to download this work and share it with others as long as you credit the authors, but you can't change the article in any way or use it commercially. More
... mation and the full terms of the licence here: https://creativecommons.org/licenses/ Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing email@example.com including the URL of the record and the reason for the withdrawal request. THE LASCAR GROUPS AND THE FIRST HOMOLOGY GROUPS IN MODEL THEORY 3 the characterization theorem (Theorem 4.4) of the first homology group and give a criterion for Lstp≡stp. We also argue that the size of the first homology group of a strong type is either 1 or ≥ 2 ℵ 0 (in Theorem 4.8, and a more detailed explanation is given in Section 6). In Section 5, we state that any connected compact abelian group can appear as the first homology group of the type of a model (Theorem 5.2), which follows from a result by Bouscaren, Lascar, Pillay, and Ziegler. We also give a more precise example of a type in a rosy theory having a non-profinite first homology group. We point out here that this paper is a result of merging two notes. The first one, single-authored by Junguk Lee, covered Section 2, Theorem 4.8 in Section 4, and Section 5.2, and the second note, jointly written by the three authors, consisted of Section 1,3, 4 and Section 5.1.