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How do I use a covering map $p \times p : \mathbb{R}\times \mathbb{R} \to S^1 \times S^1$, where $p = (\cos2\pi x, \sin2\pi x)$ to compute the fundamental group of a torus?

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You certainly know $\pi_1(S^1\times S^1)=\mathbb{Z}\times\mathbb{Z}$. This is what you want to show. First you should prove that your map gives a universal covering. Now there are at least two possibilities:

1.) Use:

If $p\times p:\mathbb{R}\times\mathbb{R}\to S^1\times S^1$ is the universal cover, then $\pi_1(S_1\times S^1)=Homeo_{S^1\times S^1}(\mathbb{R}\times \mathbb{R})$, where the last term describes the group of homeo of $\mathbb{R}\times \mathbb{R}$ commuting with $p\times p$ (not fixing a basepoint $x_0$).

This has something to do with the deck transformation group (cf. https://en.wikipedia.org/wiki/Covering_space#Deck_transformation_group.2C_regular_covers).

2.) Try to adapt the proof of the fundamental group of $S^1$ to your problem, i.e. you should prove: the map $\mathbb{Z}\times\mathbb{Z}\to\pi_1(S^1\times S^1,(1,1))$, $(m,n)\mapsto [t\mapsto(e^{2\pi im t},e^{2\pi i n t})]$ is an isomorphism of groups. The proof can be seen in Hatcher's textbook. Like in the proof you should work with the lifting property. (By the way: The loops given above are called torus knots.)

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