## Fundamental Group of S^n is trivial if n>=2

$\pi_1(S^n)=0$ if $n\geq 2$
We need the following lemma:

If a space $X$ is the union of a collection of path-connected open sets $A_\alpha$ each containing the basepoint $x_0\in X$ and if each intersection $A_\alpha\cap A_\beta$ is path-connected, then every loop in $X$ at $x_0$ is homotopic to a product of loops each of which is contained in a single $A_\alpha$.

Proof:
Take $A_1$ and $A_2$ to be the complements of two antipodal points in $S^n$. Then $S^n=A_1\cup A_2$ is the union of two open sets $A_1$ and $A_2$, each homeomorphic to $\mathbb{R}^n$ such that $A_1\cap A_2$ is homeomorphic to $S^{n-1}\times\mathbb{R}$.

Choose a basepoint $x_0$ in $A_1\cap A_2$. If $n\geq 2$ then $A_1\cap A_2$ is path-connected. By the lemma, every loop in $S^n$ based at $x_0$ is homotopic to a product of loops in $A_1$ or $A_2$. Both $\pi_1(A_1)$ and $\pi_1(A_2)$ are zero since $A_1$ and $A_2$ are homeomorphic to $\mathbb{R}^n$. Hence every loop in $S^n$ is nullhomotopic.