What is the Universal Cover of the Figure-8?

[redbowlover]

Ok so apparently the universal cover of the figure-8 can be represented by the cayley graph of the free group on two generators, discussed in Hatcher and here http://en.wikipedia.org/wiki/Rose_%28topology%29"

i can see why this is a universal cover of the figure-8. but I'm having trouble understanding why it cannot be something more simple.

for example, create a graph with one central vertex, and then four vertices surrounding it, and then connect each vertex to only the central vertex. (so you get a plus sign with vertices on the tips and one in the middle). isn't there a correct labeling on the edges of this graph to be a cover of the figure-8?

I'm not sure if this graph would be homeomorphic to the Cayley graph...ugh fractals.

 

[Hurkyl]

What is the map from your plus sign to the figure 8?

As a sanity check, have you tried lifting some test paths on the figure 8 to your plus sign? (e.g. pick a half dozen or so paths that start at the middle of the figure 8 and proceed by randomly choosing one of the four directions and winding around until it returns to the middle and doing that a few times)

 

[redbowlover]

thanks..trying to lift some paths helped me see why what i was doing didn't make sense. the map i had in mind couldn't be a covering map bc there would be no evenly covered nbhd of the vertex of the figure-8. oops :-)