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In Hatcher's book on Algebraic Topology, a lot of coverings of $S^1 \vee S^1$ has given. From these coverings, can we get different coverings of double torus (genus 2, compact surface)?

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Certainly. Hatcher outlines how to compute all covering spaces of a given space in various ways. Covering spaces of surfaces restrict to covering spaces of their 1-skeletons for any CW-decomposition. – Ryan Budney Jun 15 '10 at 17:08
You may get more responses if you put (a short version of) the question in the title. Titles on MO can be about 50% longer than tweets. Something like: "From a covering of $S^1 \vee S^1$, how can I get a covering of the two-hole torus?" – Theo Johnson-Freyd Jun 15 '10 at 22:24
Just fatten the wedge of circles to be a pretzel. His construction still gives lots of covers. Just restrict them to the boundary to get covers of your genus two surface. – Charlie Frohman Jun 16 '10 at 3:23

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