I am calculating the homology group $\#^kT^2=D_k$, and I was able to find using Mayer-Vietoris sequence, with $U=T^2-\{p\}\#^{k-1}T^2$ and $V=T^2\#^{k-1}T^2-\{p\}$, so $U=D_{k-1}$, $V=T^2$ and $U\cap V=S^1$, that $H_n(D_k)=0 $ for all $n\geq 3$. I find also that $H_0(D_k)=\mathbb Z$ because $D_k$ is path connected. So for $n=1,2$ I have in the reduced homology $$ 0\rightarrow H_2(S^1)\rightarrow H_2(D_{k-1})\oplus H_2(T^2)\rightarrow H_2(D_k)\rightarrow H_1(S_1)\rightarrow H_1(D_{k-1})\oplus H_1(T^2)\rightarrow H_1(D_k)\rightarrow H_0(S^1), $$I know that te map $H_1(S^1)\rightarrow H_1(D_{k-1})\oplus H_1(T^2)$ is the zero map, so $H_1(D_k)=\mathbb Z^{2k}$ as wished, but then I got $$ 0\rightarrow \mathbb Z^2\rightarrow H_2(D_k)\rightarrow \mathbb Z\rightarrow 0 $$ that implies $H_2(D_k)=\mathbb Z^3$, but it is supposed to be $\mathbb Z$, what am I missing?
Thanks in advance.