List of Fundamental Groups for 2-Manifolds

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


 * $\pi_1 \left({\Bbb S^1 \times \left[{0\,.\,.\,1}\right]}\right) = \Z$


 * $\pi_1 \left({\Bbb S^2}\right) = \left\{{e}\right\}$, the trivial group.


 * $\pi_1 \left({\Bbb T^2}\right) = \pi_1 \left({\Bbb S^1 \times \Bbb S^1}\right) = \Z \times \Z$


 * $\pi_1 \left({\Bbb {RP}^2}\right) = \Z_2$