Example

For $n\geqslant 1$, the sphere $\mathbf{S}_n$ is the union of two closed hemispheres, homeomorphic to the closed ball $\mathbf{B}_{n-1}$ (TG, VI, p. 12), and whose intersection is identified with the sphere $\mathbf{S}_{n-1}$. For $n\geqslant 2$, the sphere $\mathbf{S}_{n-1}$ is arcwise connected; it follows that the Poincaré group of $\mathbf{S}_n$ is trivial (cf. I, p. 127, example 3).

The sphere $\mathbf{S}_0$ has two arcwise connected components; thus one recovers that the Poincaré group of the circle $\mathbf{S}_1$ is isomorphic to a free group on one generator. More precisely, let B and C be the intersections of $\mathbf{S}_1$ with the half-planes of equations $y\geqslant 0$ and $y\leqslant 0$ in the plane $\mathbf{R}^2$. Put $a= (1,0),a'= (-1,0)$; one has $B\cap C =\{a, a'\}$; its connected components are $j_0=\{a\}$ and $j=\{a'\}$. Let $\beta$ be the class of the path $t\mapsto e^{\pi it}$ in $\mathbf{C}$; if one identifies $\mathbf{C}$ with $\mathbf{R}^2$, it joins $a$ to $a'$ in B. Analogously, let $\gamma$ be the class of the path $t\mapsto e^{-\pi it}$ joining $a$ to $a'$ in C. The path $\beta \gamma^{-1}$ is a loop at $a$, given by $t\mapsto e^{2\pi it}$. By proposition 2, its class generates the group $\pi_1(\mathbf{S}_1, a)$.