How many charts are needed to cover a 2-torus? Can you please answer this question with explanation ? I just learned about the charts needed to cover 1 and 2 spheres but got confused for the case of torus. It would be great if you guys could help.
 A: Four will do it.  Wrap one around the outside of the doughnut and another on the inside. Let them overlap a little.  Both of these are diffeomorphic to hollow cylinders, which require two patches to cover.
A: Hint $\mathbb T^2=\mathbb S^1\times\mathbb S^1$  
It is easily to see that $\mathbb S^1$ can be charted by two covers, then $\mathbb T^2$ is four.
A: If the images of the charts are not restricted to be simply connected (which they are not according to the Wikipedia definition of "chart"), then there is a two-chart-system covering the torus.
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And even with simply connected open sets, there is a way to do it with only three charts. Here is a picture showing how these charts look like on the flat torus (the right one shows that they indeed cover everything).

A: You need a minimum of 3 (contractible) charts.  Denote by T your 2-torus.  First you can cut along a set A of "figure 8" of the torus so that T\A is a topological disk/square and T\A is your 1st chart.  The figure 8 is a very small set inside T, so you can cut out another set B of “figure 8” such that T\B is also a topological disk/square such that A intersect B consists of two points and T\B is your 2nd chart.  Finally, you need a 3rd chart of disk to cover the two points.
This is a special case of Lusternik–Schnirelmann category.  In general, under some mild conditions, you can usually cover an n-dimensional manifold with (n+1)-charts.
By the way, how do I do LaTeX here?
