Cohomology groups of a torus minus a finite number of disjoint open disks

Question

I have to compute the De Rham cohomology groups of a torus with $g$ holes minus $n$ "small" (in some sense) open disks. Fix $n=g=1$ since I suppose that once this basic case is understood everything is fine. I want to use Mayer-Vietoris but I'm new to this kind of exercises and I'm not even able to identify (visualize them is the problem) two suitable open sets. Maybe it is easier to choose such sets on the "square" $Q=([0,1] \times [0,1]) \setminus D$ where $D$ is a small disk and then pass to the quotient, but am I even allowed to do so?

Any help is appreciated, thanks for your time.

Answer 1

Using $\chi(A\cup B)= \chi (A)+\chi (B)-\chi (A\cap B)$, we get

$\chi (T) =\chi (T \setminus \cup _{i=1}^n D_i) + \sum \chi (D_i) = 0$

And $\chi (D)=1$

We get $\chi (T \setminus \cup _{i=1}^n D_i)=-n$

As $b_2(T \setminus \cup _{i=1}^n D_i)=0$, we get $1-b_1((T \setminus \cup _{i=1}^n D_i))=-n$

$b_1(T \setminus \cup _{i=1}^n D_i)=n+1$

The coefficients are taken where you want, $\bf Z$ for instance.