The exercise looks like this:
- Prove that the $\mathbb R/\mathbb Z$ and $S^1$ are diffeomorphic.
- Prove that $\mathbb R^n/\mathbb Z^n$ and $(\mathbb R/\mathbb Z)^n$ are diffeomorphic.
- Deduce that $\mathbb T^n$ is diffeomorphic to $(S^1)^n $.
What I did so far:
I took $\phi:$$ \mathbb R \to S^1 $ where $\phi(x)=(cos(2\pi x),sin(2\pi x))$.
$\pi$ :$\mathbb R \to \mathbb R/\mathbb Z\ $ a surjective map
and then $\psi$ : $ \mathbb R/\mathbb Z \to S^1 $.
So $\phi$ is a composition of these two maps.... I thought if I prove that the composition is a diffeomorphism it'll be the answer!
Thank you.
You have too much in this question: even just the case $n=1$ needs work, so I'll give you a critique of the $n=1$ case. The main problem with your approach is lack of rigor regarding quotient maps and quotient topologies.
First, regarding the function $\pi : \mathbb R \to \mathbb R / \mathbb Z$ you need not just that it is surjective, but you need that it is a quotient map.
Second, you need to apply the universal property of quotient maps to the composition formula $\phi = \psi \circ \pi$; from that, you can deduce that $\psi$ is also a quotient map. (You should also correct your description/formula for $\phi$: its domain is $\mathbb R$, not $\mathbb R / \mathbb Z$).
Third, you need to prove that $\psi$ is one-to-one, and from that you can finally deduce that $\psi$ is a homeomorphism.
Here's just a hint for the general case: proceed similarly, starting from $$\phi(x_1,...,x_n) = \biggl(\bigl(\cos(2\pi x_1),\sin(2 \pi x_1)\bigr),...,\bigl((\cos(2\pi x_n),\sin(2\pi x_n)\bigr)\biggr) $$