3
$\begingroup$

The universal cover of the torus $T$ is the complex plane $\mathbb{C}$. If $p: \mathbb{C} \to T$ is the covering map, why is $p$ doubly periodic?

  • 1
    Universal covering map is unique up to an automorphism of the covering space. A complex-analytic automorphism of $\mathbb C$ is of the form $az+b$, hence preserves double-periodicity (though it may change the periods themselves). So, as long as we have one doubly-periodic cover such as the quotient map, we know that all covering maps are doubly-periodic.2012-06-10

1 Answers 1

1

Since $\mathbb C$ is simply connected, it is a universal covering space. Any two covering maps $\mathbb C\to\mathbb T$ are related by an automorphism of $\mathbb C$. Such automorphisms are linear, therefore preserve periodicity. So if one covering map is doubly periodic, all of them are. To get one such map, use the quotient map that comes from the definition of torus as the quotient of $\mathbb R^2$ by a lattice.