List of Fundamental Groups for 2-Manifolds

Theorem

This article needs to be linked to other articles. In particular: two-mainfold, also review links for everything else because it's all over the place You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

For the following two-manifolds, the fundamental group for any point in $X$, written $\map {\pi_1} X$ is isomorphic to the listed group:

$\map {\pi_1} {\Bbb S^1 \times \closedint 0 1} = \Z$

$\map {\pi_1} {\Bbb S^2} = \set e$, the trivial group.

$\map {\pi_1} {\Bbb T^2} = \map {\pi_1} {\Bbb S^1 \times \Bbb S^1} = \Z \times \Z$

$\map {\pi_1} {\Bbb {RP}^2} = \Z_2$

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.