What would be an example of a nontrivial, i.e. multi-sheeted, covering of the torus? I would greatly appreciate any help that you could give me.
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2Can you find a non-trivial covering of the circle $S^1$? – Mariano Suárez-Álvarez Jun 10 '13 at 18:36
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well yes, but how does that help? – picklekong Jun 10 '13 at 18:38
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1The torus is $S^1 \times S^1$. – Qiaochu Yuan Jun 10 '13 at 18:43
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2Quick note: the cover will be a torus (besides the universal cover), since for an n-fold cover $A\to B$ we have $\chi(A)=n\chi(B)$, and here we have $\chi(B)=0$. – Chris Gerig Jun 10 '13 at 19:03
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The torus is $\cong \mathbb R^2/\mathbb Z^2$. Let $\Lambda$ be any proper sublattice of $\mathbb Z^2$ (for example the one spanned by $(3,4)$ and $(8,-6)$), then $\mathbb R^2/\Lambda$ is what you are looking for.
Hagen von Eitzen
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2Well, strictly he is looking for a map, not another construction for the torus :-) – Mariano Suárez-Álvarez Jun 10 '13 at 18:58