The Lascar groups and the 1st homology groups in model theory

TL;DR

This paper explores the relationship between Lascar groups and first homology groups in model theory, establishing canonical maps, independence from certain choices, and implications for the structure of types.

Contribution

It introduces a canonical surjective homomorphism from the Lascar group to the first homology group and characterizes its kernel, showing independence from the choice of independence relation.

Findings

The first homology group is at least continuum-sized unless trivial.

A criterion for equality of strong and Lascar types is provided.

Any abelian connected compact group can be realized as a first homology group.

Abstract

Let $p$ be a strong type of an algebraically closed tuple over $B=\acl^{\eq}(B)$ in any theory $T$. Depending on a ternary relation $\indo^*$ satisfying some basic axioms (there is at least one such, namely the trivial independence in $T$), the first homology group $H^*_1(p)$ can be introduced, similarly to \cite{GKK1}. We show that there is a canonical surjective homomorphism from the Lascar group over $B$ to $H^*_1(p)$. We also notice that the map factors naturally via a surjection from the `relativised' Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of $p$ is independent from the choice of $\indo^*$, and can be written simply as $H_1(p)$. As consequences, in any $T$, we show that $|H_1(p)|\geq 2^{\aleph_0}$ unless $H_1(p)$ is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group. We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.