Mayer–Vietoris computation of first homology of the torus

Choose two open arcs $A,B\subset S^1$ covering the first factor with $A\cap B=I_0\bigsqcup I_1$, two open arcs. Put $U=A\times S^1$ and $V=B\times S^1$. Each overlap component and each of $U,V$ retracts onto the second factor. After ordering $I_0,I_1$, the Mayer--Vietoris map $H_1(U\cap V;G)\to H_1(U;G)\oplus H_1(V;G)$ is $(a,b)\mapsto (a+b,-a-b)$, since inclusion preserves the second-factor circle and the sequence uses opposite signs for $U$ and $V$. The same formula holds in degree zero. Its kernel is $\{(g,-g):g\in G\}$ and its cokernel is $G$, with no freeness assumption on $G$.

Exactness gives $0\to G\to H_1(T^2;G)\stackrel{\delta }{\to }G\to 0$, where the right $G$ is the displayed degree-zero kernel. Fix $y_0\in S^1$ and orient $S^1\times \{y_0\}$ so that its arc in $U$ has boundary $g([x_0]-[x_1])$, with $x_j\in I_j$. Splitting its singular cycle into the arcs in $U$ and $V$, the chain-level Mayer--Vietoris connecting map sends its class with coefficient $g$ to $(g,-g)$. Hence this first-factor circle supplies a homomorphic section of $\delta$ for every abelian $G$. The second-factor circle at $x_0$ maps to $(g,0)$ in $H_1(U)\oplus H_1(V)$ and represents the cokernel generator of step 1.1. These two circle maps identify $H_1(T^2;G)$ with $G\oplus G$.