The Lascar groups and the first homology groups in model theory

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 ⌣ | * 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 [3]. 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 ⌣ | * , and can be written simply as H 1 (p).As consequences, in any T , we show that |H 1 (p)| ≥ 2 ℵ0 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.In this paper we study the first homology group of a strong type in any theory.Originally, in [3] and [4], a homology theory only for rosy theories is developed. Namely, given a strong type p in a rosy theory T , the notion of the nth homology group H n (p) depending on thorn-forking independence relation is introduced. Although the homology groups are defined analogously as in singular homology theory in algebraic topology, the (n + 1)th homology group for n > 0 in the rosy theory context has to do with the nth homology group in algebraic topology. For example as in [3], H 2 (p) in stable theories has to do with the fundamental group in topology. 1 2 JAN DOBROWOLSKI, BYUNGHAN KIM, AND JUNGUK LEE theories, H 1 (p) is detecting somewhat endemic properties of p existing only in model theory context.Indeed, in every known rosy example, H n (p) for n ≥ 2 is a profinite abelian group. In [5], it is proved to be so when T is stable under a canonical condition, and conversely, every profinite abelian group can arise in this form. On the other hand, we show in this paper that the first homology groups appear to have distinct features as follows.Let p = tp(a/B) be a strong type over B = acl eq (B) in any theory T . Fix a ternary invariant independence relation ⌣ | * among small sets satisfying finite character, normality, symmetry, transitivity and extension. (There is at least one such relation, by putting A ⌣ | C D for any sets A, C, D.) Then we can analogously define H * 1 (p) depending on ⌣ | * , (which of course is the same as H 1 (p) when ⌣ | * is thorn-independence in rosy T ). In this note, a canonical epimorphism from the Lascar group over B of T to H * 1 (p) is constructed. Indeed, we also introduce the notion of the relativised Lascar group of a type which is proved to be independent from the choice of the monster model of T , and the homomorphism factors th...

The Relativized Lascar Groups, Type-Amalgamation, and Algebraicity

“…This paper continues the study started in [3], where a connection between the relativized Lascar groups of a strong type and the first homology group of the strong type was established. If T is G-compact (for example when T is simple), then the relativized Lascar group of a strong type is compact and connected, and we use compact group theory to obtain results presented here.…”

“…The Lascar group only depends on the theory and it is a quasi-compact topological group with respect to a quotient topology of a certain Stone type space over a model ( [2] or [13]). More recently, the notions of the relativized Lascar groups were introduced in [3] (and studied also in [11] in the context of topological dynamics). Namely, given a type-definable set X in a large saturated model of the theory T, we consider the group of automorphisms restricted to the set X quotiented by the group of restricted automorphisms fixing the Lascar types of the sequences from X of length .…”

“…We will give an example (in Example 2.3) where Gal 1 L (p) and Gal 2 L (p) are distinct. In [3,Remark 3.4], a canonical topology on each of the above groups was defined. With these topologies, they become quotients of the topological group Gal L (T ).…”

“…Fact 0.4 [3]. Autf (p) = Autf fix (p), and, for each (≤ ), Gal L (p) does not depend on the choice of a monster model, and is a quasi-compact connected topological group.…”

Boundedness and absoluteness of some dynamical invariants in model theory

“…Let p 1 = tp(ᾱ/∅). Following the notation from [4], we define Gal fix,1 L (p 1 ) as the quotient of the group of all elementary permutations of p 1 (C) by the subgroup Autf fix,1 L (p 1 (C)) of all elementary permutations of p 1 (C) fixing setwise the E L -class of each realization of p 1 (but not necessarily of each tuple of realizations of p 1 , and that is why we write superscript 1). Gal fix,1 L (p 1 ) can be, of course, identified with the quotient of Aut(C) by the group Autf fix,1 L,p 1 (C) of all automorphisms fixing setwise the E L -class of each realization of p 1 .…”