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Irrational Winding of the Torus

  1. Apr 24, 2012 #1

    jgens

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

    I am having difficulty however showing that [itex]\pi(L)[/itex] is not locally Euclidean. From drawing a few pictures, I think every neighborhood of [itex]\pi(0)[/itex] is disconnected (which would be enough to complete the proof), but I am having difficulty showing this. Any help?
     
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  3. Apr 24, 2012 #2

    quasar987

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    Look at p.158 of the book of John Lee.
     
  4. Apr 24, 2012 #3

    jgens

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