This is a problem on a review for some upcoming quals:
Prove there are uncountably many 3-dimensional foliations on a 5-dimensional torus.
Unless I am looking at this wrong it seems like a pretty straight forward intuitive sort of problem. My mind first went to just the plain old torus $\mathbb{R}^2/\mathbb{Z}^2$. There are uncountably many 1-dimensional foliations. This can be seen by fixing a slope and taking lines through every point with the given slope. If the slope is rational the lines will eventually double back on themselves and form a loop. If they are irrational they will go on forever in both directions.
Similarily, for $\mathbb{R}^3/\mathbb{Z}^3$ pick a vector and then at every point pick a plane going through that point tangent to the vector. This gives uncountably many foliations of dimension 2.
For the problem given I suspect it's the same idea. In $\mathbb{R}^5$ there are uncountably many 3 dimensional planes. Line them all up facing the same direction, and in the quotient you will have a 3-dimensional foliation of $\mathbb{R}^5/\mathbb{Z}^5$.
I'm wondering if this is the way I want to look at this problem, or if there is a better way to view it.
Thanks :)