The Lascar groups and the first homology groups in model theory

Let p be a strong type of an algebraically closed tuple over B = acl(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 H1(p). As consequences, in any T , we show that |H1(p)| ≥ 2א0 unless H1(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 Hn(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], H2(p) in stable theories has to do with the fundamental group in topology. This implies that, already in rosy theories, H1(p) is detecting somewhat endemic properties of p existing only in model theory context. All authors were supported by Samsung Science Technology Foundation under Project Number SSTF-BA1301-03. The third author was also supported by NRF of Korea grant 2013R1A1A2073702.