List of Fundamental Groups for 2-Manifolds

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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.


$\pi_1 \left({\Bbb T^2}\right) = \pi_1 \left({\Bbb S^1 \times \Bbb S^1}\right) = \Z \times \Z$
$\pi_1 \left({\Bbb {RP}^2}\right) = \Z_2$


Proof