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I have an assignment where I have to list all $2$ and $3-$fold covers for the figure $8$ graph and I am having some difficulties on how to think about it.

As far as the degree $2$ are concerned, first. Is there a way of intuitively picture the covering spaces for the graph? I know that the central vertex will have to be image of two vertices, and this helps. How do I guess all of the covers with this property now?

Any suggestions are appreciated! Thank you very much

  • 1st try to see that how many edges and vertices should be there in a 2 or 3 fold covering, and then try to figure it out that among them which could be your required covering space – Anubhav Mukherjee Oct 30 '15 at 12:20
  • ok so my figure eight has 1 vertex and 2 edges, right? So I would need to double it for a 2 fold cover, getting 2 vertices and 4 edges –  Oct 30 '15 at 12:38
  • do you know how many of each there are, up to isomorphism? –  Oct 31 '15 at 19:35

1 Answers1

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As you have written in your comment in the case of the 2-fold cover you will have two vertices with neighbourhoods mapping homeomorphically to the one downstairs so in the "lifted" you also 4 have small arcs mapping to the 4 arcs in the base vertex.

I would say to try and label the two loops in the figure 8 graph and fix an orientation. Then try to look at all the possible ways of connecting the "lifted" vertices by respecting the orientations.

Spotty
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  • Thank you for your help. How many 2-fold covers for the graph are there? I have three different, but my professor said there were four. And what about the 3-fold covers? I found there were seven, I don't seem to know how to prove that they are not isomorphic and why there are no more... –  Oct 31 '15 at 19:29
  • for the 2-fold cover did you count the disconnected one? – Spotty Oct 31 '15 at 20:38
  • only the connected ones. that's maybe why he said there were 4 –  Oct 31 '15 at 20:40
  • there are 3 connected ones and a disconected one – Spotty Oct 31 '15 at 20:42
  • yes so I have them all. and what about the $3$-fold? –  Oct 31 '15 at 20:42
  • There are 7 3-fold covers, to show they are not isomorhic you might try to look at the action of $\pi_1$. In general these covers are classified by conjugacy classes of maps $\pi_1(X)\mapsto\Sigma_n$ which will tell you how many there are, you can read about this in Hatcher when he talks about the covering group actions – Spotty Oct 31 '15 at 20:46
  • I am not able to use that tool yet! But thank you for your help. –  Oct 31 '15 at 20:48
  • What about computing deck groups for each of the covering? Trying to sketch it for each cover. I just think of the possible combination of edges? –  Oct 31 '15 at 20:50