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:


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


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


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


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