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almost there

Member

Registered: 2009-11-11

Posts: 21

Topology: Covers of figure-eight space

I have to find all covers of the figure-eight space where the preimage of each point has exactly 3 distinct elements in the covering space.

I am really confused on this. I know that one such covering space is 8 X {1, 2, 3}, where "8" denotes the figure-eight space. This is just 3 copies of the figure-eight. But what other ones are there? How do I "find" them? I am not even sure how to begin constructing the others, whatever they may be. Any advice would be priceless!

Thank you in advance.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Topology: Covers of figure-eight space

Have you heard of a Universal cover?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

almost there

Member

Registered: 2009-11-11

Posts: 21

Re: Topology: Covers of figure-eight space

Nope, afraid not. I'd be interested in knowing how that could help, but as that concept hasn't entered in lecture yet, my professor must have solutions in mind that don't use a universal cover.

I'm still working on this, but so far I've managed to doodle what I believe to be some 3-fold covers of the figure-eight space. I'm supposed to find all 3-fold covers of the space though. How can I find out whether I have them all? Or maybe some of mine are redundant. I think they are all distinct, but its all based on intuition. So I guess I'd also like to know how to tell whether two of my doodles represent the same cover space. Advice?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Topology: Covers of figure-eight space

It sounds like you don't have many tools, for example the correspondence between covering spaces and subgroups of the fundamental group.  If that's the case, then I don't know how you'd ever prove you have all.  Think about the behavior at the center point of the figure 8.  In a neighborhood around the preimage of this point, your cover must at the same way.  All other points are uninteresting because lines will lift to lines (remember, you're looking locally).

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

almost there

Member

Registered: 2009-11-11

Posts: 21

Re: Topology: Covers of figure-eight space

Thanks--your explanation confirms what I inferred as I was trying to finagle a way to draw some of the covering spaces, and also bolsters my confidence in what covering spaces I did come up with. Do you happen to know how many 3-fold covering spaces of the figure-eight space there are?

I'm sure that we'll get to it soon, but out of curiosity what is the correspondence you speak of? I know I could look it up, but me and course books have rarely agreed, and the book for this course is no exception. I mean, its nice mathematically, but its got its front cover too far up its index to be pedagogically effective (as most do).

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Topology: Covers of figure-eight space

I'm counting 6, and I believe my enumeration covers all cases (up to equivalence).

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

uc_david

Guest

Re: Topology: Covers of figure-eight space

I believe that there are 7 non-equivalent triple covers of the figure eight space.