----- > [!proposition] Proposition. ([[punctured Euclidean space and the unit sphere have isomorphic fundamental groups]]) > The [[inclusion map]] $\iota:\mathbb{S}^{n} \to \mathbb{R}^{n+1}-\b 0$ [[homomorphism of fundamental groups induced by a continuous map|induces]] a [[group isomorphism|isomorphism]] between the [[fundamental group|fundamental groups]] $\pi_{1}(\mathbb{S}^{n},x_{0})$ and $\pi_{1}(\mathbb{R}^{n+1}-\b 0)$. > [!proof]- Proof. ([[punctured Euclidean space and the unit sphere have isomorphic fundamental groups]]) > The case $n=1$ appeared earlier as part of a homework problem. [[fundamental group#^ca2f57|See here]]. Now we let $n$ be any natural number. > Set $X=\mathbb{R}^{n+1}- \b 0$ and $v_{0}=(1,0, \dots, 0)$. Let $f:X \to \mathbb{S}^{n}$ be the normalization map $v \xmapsto{f} \frac{v}{\|v\|_{2}}$. Clearly $f \circ \iota =\id$ on $\mathbb{S}^{n}$. In general $\iota \circ f:X \to X$ is not $\id$, but the maps *are* [[homotopy|homotopic]]. Indeed, the [[straight-line homotopy]] $H:X \times I \to X$ given by $H(x,t)=(1-t)x+ tx / \|x\|$ is a [[homotopy]] between $\id$ and $\iota \circ f$ (since $H$ is never $\b 0$ we can just invoke the general results regarding [[straight-line homotopy]] for [[convex set]]s). > The rest of the proof is really no different than the case $n=1$. ----- #### ---- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM outgoing([[]]) FLATTEN file.tags GROUP BY file.tags as Tag