Is π(L) a Submanifold of the Torus?

[jgens]

I am trying to prove the following result: Fix a,b \in \mathbb{R} with a \neq 0. Let L = \{(x,y) \in \mathbb{R}^2:ax+by = 0\} and let \pi:\mathbb{R}^2 \rightarrow \mathbb{T}^2 be the canonical projection map. If \frac{b}{a} \notin \mathbb{Q}, then \pi(L) (with the subspace topology) is not a submanifold of \mathbb{T}^2.

I am having difficulty however showing that \pi(L) is not locally Euclidean. From drawing a few pictures, I think every neighborhood of \pi(0) is disconnected (which would be enough to complete the proof), but I am having difficulty showing this. Any help?

 

[quasar987]

Look at p.158 of the book of John Lee.

 

[jgens]

Look at p.158 of the book of John Lee.

Thanks! I (finally) figured out a brute force method using the Hurwitz Theorem that works, but Lee's solution is much cleaner.