Finding the de Rham Cohomology group of $\mathbb{S}^2\backslash \{a,b,c\}$. I am trying to understand how to compute the de Rham cohomology group. Could someone show me how I would find the cohomology group of $\mathbb{S}^2\backslash\{n\text{ points of } \mathbb{S}^2 \}$. I know I should find the de Rham cohomology group for $n=1,2,3,4,..$ till I see a pattern. However, I have little experience doing this and can't find a good example I could follow. Could someone show me how to compute these for several n-values?
 A: Since $\mathbb S^2\setminus\{\text{point}\}$ is homeomorphic to $\mathbb R^2$, we focus on the question of finding the deRham cohomology group
$$H^1(\mathbb R^2 \setminus\{p_1, \cdots, p_n\})$$
for all $n$ and $i=1,2$. For simplicity write $X_n  = \mathbb R^2 \setminus\{p_1, \cdots, p_n\}$. We show by induction that 

$H^2(X_n) = 0$, while $H^1(X_n) \cong \mathbb R^n$.

The case $n=0$ is trivial (Poincare lemma). Assume the above for some $n-1$. By a diffeomorphism, one may assume that $p_i = (i,0) \in \mathbb R^2$. 
Let $X_n^+ = X_{n-1} \setminus\{ (i, t): t\ge 0\}$ and  $X_n^- = X_{n-1} \setminus\{ (i, t): t\le 0\}$. Then 
$$ X_n^+ \cup X_n^- = X_n, \ \ X_n^+ \cap X_n^- \cong X_{n-1} \cup R,$$
where $R = (p_n,\infty) \times \mathbb R$. Using (part of) the MV sequence, which is 
$$\to H^1(X_n^+) \oplus H^1(X_n^-)  \overset{g}{\to}  H^1(X_{n-1} \cup R)  \to H^2(X_n) \to 0$$
Note that $X_n^\pm \cong X_{n-1}$ and $g$ is given by $g(a, b) = a-b$ (Note $H^1(X_{n-1} \cup R) = H^1(X_{n-1})$). This shows that $H^2(X_n) = 0$ as $g$ is surjective. On the other hand, using
$$H^0(X_n^+)\oplus H^0(X_n^-)\to H^0(X_{n-1} \cup R)\overset{d^*}{\to} H^1(X_n) \to H^1(X_n^+) \oplus H^1(X_n^-) \to $$
The first two groups are both $\mathbb R^2$ and the maps between them is $(s, t) \mapsto (s-t,s-t)$. Together with the expression for $g$, we see that $H^1(X_n)$ is $n$-dimensional. Thus the assertion is proved by induction.
