Remark 3

To the exact sequence $0\rightarrow \Gamma (S)\rightarrow L(S)\longrightarrow^{exp_S}S\rightarrow 0$ is associated an isomorphism from $\Gamma (S)$ to the fundamental group of S, called canonical in the sequel. For any morphism of tori $f: S\rightarrow S',\Gamma (f)$ can then be identified via the canonical isomorphisms $\Gamma (S)\rightarrow \pi_1(S)$ and $\Gamma (S')\rightarrow \pi_1(S')$ with the homomorphism $\pi_1(f) :\pi_1(S)\rightarrow \pi_1(S')$ induced by $f$. Note that this gives another interpretation of the exact sequence (5) (cf. General Topology, Chap. XI, in preparation).