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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 trivialindependence in $T$), the first homology group $H^*_1(p)$ can be introduced,similarly to \cite{GKK1}. We show that there is a canonical surjectivehomomorphism from the Lascar group over $B$ to $H^*_1(p)$. We also notice thatthe map factors naturally via a surjection from the `relativised' Lascar groupof 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. Dueto this characterization, it follows that the first homology group of $p$ isindependent 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 andLstp of algebraically closed tuples using the notions of the first homologygroup and a relativised Lascar group. We also argue how any abelian connectedcompact group can appear as the first homology group of the type of a model.

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