By using the Mayer-Vietoris sequence in reduced homology :
I have to calculate the homology groups of :
The torus $\mathbb{T}^2 :=[0;1]^2 /\mathcal{T}$ by using the following decomposition $X_1 := \mathbb{T}^{2} -\{(1/2;1/2)\}$, $X_2 = ]0,1[^{2}$ et $A=X_1 \cap X_2$.
$\mathcal{T}$ is an equivalence relation meaning that we have identified the boudaries of the square $[0;1]^2$ : $(1,t)\sim (0,t)$ and $(t,1)\sim (t,0)$;$t\in[0;1]$. The point $(1/2;1/2)$ is the center of the square.
I have already done this work : the rose with 2 petals is a retract by deformation of $X_1$ so $\tilde{H}_0(X_1)=\mathbb{Z}$ $\tilde{H}_1(X_1)=\mathbb{Z\oplus\mathbb Z}$ and $\tilde{H}_n(X_1)=0$ for $n\geq 2$
$\tilde{H}_n(X_2)=0$ for all $n$.
But i am stuck here. How can i compute $H_1(\mathbb T^2)$ and $H_2(\mathbb T^2)$ ? Thanks for any help.