TITLE: Van Kampen theorem to prove all spheres have trivial fundamental group
QUESTION [7 upvotes]: I'm trying to prove that $\pi_1(S^n)=0$  when $n>1$.
I've noticed that if you just take the north pole and south pole from each sphere, you can apply Van Kampen theorem. This doesn't fail as when $n>1$ it's still simply connected as it's homotopic to a circle. 
But, how would you make this argument more formal? Would you use induction. But, can't see what the induction step will be. Help?

REPLY [12 votes]: As you said you have to use Van Kampen theorem, but you don't need any induction argument. Let's consider a pair of antipodal point $P,Q \in S^n$ and consider the open sets  $A=S^n \setminus \{P\}$ and $B=S^n \setminus \{Q\}$. The opensets $A$ and $B$ are both homemorphic to $D^n$ (the $n$-dimensional disk) and so are both simply connected, their intersection is path-connected, so by Van Kampen theorem $\pi_1(S^n)$ is quotient of $\pi_1(A)*\pi_1(B)=0*0=0$ (here $*$ is the free product of groups), thus $\pi_1(S^n)=0$.
Clearly this argument holds only for $n>1$ because just in this case the intersection of the open sets $A$ and $B$ is path-connected.