What is the Universal Cover of the Figure-8?

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In summary, the universal cover of the figure-8 can be represented by the cayley graph of the free group on two generators, as discussed in Hatcher and Wikipedia. While there are simpler graph representations, such as a plus sign with vertices connected to a central vertex, these do not accurately map to the figure-8 and therefore cannot be a covering map. Lifting paths on the figure-8 helped to demonstrate this.
  • #1
redbowlover
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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" [Broken]

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.
 
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  • #2
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)
 
  • #3
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 :-)
 

1. What is the Universal Cover of Figure-8?

The Universal Cover of Figure-8 is a topological space that covers the figure-8 space, meaning that the figure-8 space can be mapped onto the Universal Cover in a continuous and one-to-one manner.

2. How is the Universal Cover of Figure-8 related to the figure-8 space?

The Universal Cover of Figure-8 is a covering space of the figure-8 space. This means that it is a topological space that maps onto the figure-8 space in a continuous and one-to-one manner.

3. Why is the Universal Cover of Figure-8 important in mathematics?

The Universal Cover of Figure-8 plays an important role in algebraic topology, as it helps to understand the fundamental group of the figure-8 space. It also has applications in other areas of mathematics, such as knot theory and graph theory.

4. How is the Universal Cover of Figure-8 constructed?

The Universal Cover of Figure-8 can be constructed by taking an infinite number of copies of the figure-8 space and gluing them together in a specific way. This process is known as the covering transformation.

5. Can the Universal Cover of Figure-8 be visualized?

While the Universal Cover of Figure-8 cannot be visualized in its entirety, it can be represented by an infinite branching tree-like structure known as a Cayley graph. This helps to better understand the relationship between the Universal Cover and the figure-8 space.

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