---- > [!theorem] Theorem. ([[past one dimension, the sphere is simply connected]]) > The [[sphere]] $\mathbb{S}^{k-1} \subset \mathbb{R}^{k}$ is [[simply connected]] when $k \geq 2$. > [!note] Remark. > This is a consequence of [[weak Seifert-van Kampen theorem|weak SVK]] and [[past one dimension, punctured euclidean space is path-connected]]. > [!proof]- Proof. ([[past one dimension, the sphere is simply connected]]) > [[stereographic projection#^68968c|Using stereographic projection]] we can cover $\mathbb{S}^{k-1}$ with two [[coordinate patch|patches]], one of which covers $\mathbb{S}^{k-1} - \{ N \}$ and the other which covers $\mathbb{S}^{k-1}-\{ S \}$. Hence by [[TODO|TODO: result about patch domains homeomorphic to all of R^n]] we have $\mathbb{S}^{k-1} - \{ N \}$ and $\mathbb{S}^{k-1}-\{ S \}$ are each [[homeomorphism|homeomorphic]] to $\mathbb{R}^{k-1}$; each set is therefore [[simply connected]]. Moreover, their intersection is [[path-connected]], for it is [[homeomorphism]] to $\mathbb{R}^{k-1}-\{ \b 0 \}$ and [[past one dimension, punctured euclidean space is path-connected]]. Thus [[weak Seifert-van Kampen theorem|weak SVK]] applies to tell us that the [[fundamental group]] of $\mathbb{S}^{k-1}$ is [[generating set of a group|generated by]] two trivial elements... it is hence trivial. ---- #### ----- #### 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