# Irrational Winding of the Torus

1. Apr 24, 2012

### 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?

2. Apr 24, 2012

### quasar987

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

3. Apr 24, 2012

### jgens

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