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I'm trying to complete the following exercise I found in a topology book: Construct a space A which is path-connected with fundamental group equal to $\langle r,s | r^2 s^3 r^3 s^2 = 1\rangle$, and find a unique (connected) double-covering B. Find some presentation also for $\Pi_1(B)$.

Now the space I had in mind for A was a 10-sided polygon with edges identified, much like they are in (say) the Klein bottle, the first 2 edges identified in the same direction, and also identified with the 6th, 7th and 8th edges in that direction, and the 3rd, 4th, 5th, 10th and 10th edges identified all in the same direction too. Then by Seifert Van-Kampen we can remove a small disc in the center of the polygon to get the fundamental gp we want. However, I am having problems picturing any sort of sensible double-cover for A: could anyone suggest anything? For something like the Klein bottle it's reasonably easy to just join another Klein bottle on at one of the edges to get a torus, but can we do the same here? Just add another copy of A associated at 1 edge (or all edges)? I'm not entirely sure how it works.

I have tagged this as homework because it is an exercise, I have no-one to hand it in to if that's relevant. Thanks! -Roy

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Hi Roy, welcome to this site! I removed the homework tag since it should be used for actual assigned homework. –  t.b. Jun 4 '11 at 21:26

1 Answer 1

Let $G = \langle r,s | r^2 s^3 r^3 s^2 = 1\rangle$, the fundamental group of $A$.

By the Galois correspondence, the fundamental group of $B$ should be isomorphic to an index-two subgroup $H$ of $G$. Any subgroup of index two is normal, so $H$ is the kernel of some surjective homomorphism $\varphi\colon G \to \mathbb{Z}/2\mathbb{Z}$. From the relation $r^2 s^3 r^3 s^2 =1$, we can see that $5 \varphi(r) + 5 \varphi(s) = 0$ in $\mathbb{Z}/2\mathbb{Z}$, so the only possibility is $\varphi(r) = \varphi(s) = 1$. We conclude that neither $r$ nor $s$ lifts to a loop in the cover.

So the 1-skeleton of the double cover has two vertices and four edges, like this:

enter image description here

The single 2-cell in $A$ lifts to a pair of 2-cells in $B$, which are attached as follows:

enter image description here

You can now use this cell complex to get a presentation for the fundamental group of $B$.

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