List of Fundamental Groups for 2-Manifolds

= Theorem =

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

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

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

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

$$\pi_1(\mathbb{RP}^2) = \mathbb{Z}_2$$

= Proof =