Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I know the covering spaces of the of a torus $T^2$ are homeomorphic to $T^2,S^1\times\mathbb{R},\mathbb{R}^2$. I am interested in finding all of the covers with covering space $T^2$. The subgroups of $\pi_1(T^2)=\mathbb{Z}\times\mathbb{Z}$ with covering space $T^2$ are of the form $\langle (a,b),(c,d)\rangle\cong\mathbb{Z}\times\mathbb{Z}$.

The cover for $n\mathbb{Z}\times m\mathbb{Z}$ is $p:T^2\rightarrow T^2, p(x,y)=(x^n,x^m)$. How can I generalize this to arbitrary 2 dimensional subgroups.

share|cite|improve this question

If the subgroup is spanned by two linearly independent $(a,b)$ and $(c,d)$, then the cover is $p(x,y) = (x^a y^b, x^c y^d)$. This map is surjective because of the independence of the spanning vectors.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.