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.

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.


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