Fundamental group of the circle. Uses Homotopy lifting lemma and corollaries (see Covering).

$\pi(S^1, 1) \cong \mathbb{Z}$. which are the Rotation numbers

Fundamental group of the torus. $\mathbb{Z}^2$

Real projective space (see Projective plane). its topology. Fundamental group is isomorphic to $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$. Shown using Path lifting lemma and Homotopy lifting lemma

See discusion here about the donut-mug: https://www.facebook.com/photo.php?fbid=10213148857979196&set=gm.2018441631801224&type=3&theater and its homotopy groups