Cohomology group of a torus with g holes I have to compute the cohomology groups of a torus with g holes (the Riemann surface of genus g).
first I have computed the cohomology of a Torus with 3 holes in the following way:
I pick a covering $\{U,V \}$ so that $U,V$ look like a pants, cutting suitably the torus, and then by Mayer-Vietoris I computed the cohomology group.
But how I can generalize it to a torus with g holes?
The exercice continues asking:


*

*Compute  the cohomology groups of a torus with g holes minus n point?

*Compute  the cohomology groups of a torus with g holes minus n small open disc pairwise disjoint.


How could I proceed with this two point?
 A: I suggest you next compute the cohomology of a torus minus two points, and a cylinder. Let me draw a schematic picture of a 3-holed torus:
()()()
Now I can make a cut across the 4th hole:
()()(|)
and the result, on the right, is a cylinder; on the left, by retracting the "arms" of the rightmost parenthesis, I get a 2-hole torus minus two points. 
If you knew the cohomology of the twice-punctured 2-holed torus, you could use MV to find the 3-holed torus cohomology. 
So the idea here is to use induction: 


*

*Find cohomology of a torus and a twice-punctured torus. 

*Find cohomology of a cylinder.

*Use MV to find cohomology of 2-holed torus

*Carefully figure out cohomoloyg of twice-punctured 2-holed torus

*Use MV to find cohomology of 3-holed torus

*Carefully figure out cohomology of twice-punctured 3-holed torus

*...


For the twice-punctured 2-holed torus, you can also use MV: take the 2-holed torus $S$ with points $P$ and $Q$ to be removed. Let $A$ and $B$ be small disks containing $P$ and $Q$ repectively. Let $U = S - \{P, Q\}$, $V = A \cup B$, and apply MV to $(S, U, V)$. 
(The same reasoning applies to a many-punctured k-holed torus. )
