# The covering space of a bouquet of 2 circles corresponding to a normal subgroup

Consider $S_3$ with this presentation: $S_3=\left\langle\sigma,\tau:\sigma^2=1, \sigma\tau=\tau^{-1}\sigma\right\rangle$. Let F be the free group with two generators $s,t$ and $R$ the minimal normal subgroup of $F$ containing $s^2$ and $sts^{-1}t$. What is the covering space of the bouquet of 2 circles corresponding to $R$?

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## 1 Answer

Since you've recognized $F/R$ as isomorphic to $S_3$, you already know how large that covering space is, meaning what is the order of the covering (what is it?), and you also know what permutations define the paths that lie over each of the circles in the bouquet (they correspond to $\sigma$ and $\tau$). Can you wrap this up from here?

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actually I'm not very good in algebraic topology, could you give me a precise answer, please? – Berry Sep 25 '11 at 5:16
It's not going to be easy nor useful to explain without knowing something about the context of the question. What do you already know about covering spaces, fundamental groups, subgroups and how the correspond to each other? Is this a homework problem? – Alon Amit Sep 25 '11 at 5:56
I studied all these stuff years ago and now I don't remember it. This is a part of an exercise on cohomology of groups. But however try to explain, I will read all that theory again to understand well your question – Berry Sep 25 '11 at 6:10
ok I got it!!!! – Berry Sep 27 '11 at 6:47