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Is there any good guide on covering space for idiots? Like a really dumped down approach to it . As I have an exam on this, but don't understand it and it's like 1/6th of the exam.

So I'm doing Hatcher problem and stuck on 4.

  1. Construct a simply-connected covering space of the space $X \subset \mathbb{R}^3$ that is a union of a sphere and diameter.

All I can think of is just connecting a bunch of spheres in a line.

But, yeah pretty scared will fail my degree because of this. So I need a good guide of covering spaces that isn't Hatcher. The only other uses heavy category theory which is even worse to read.

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What about Munkres' "Topology" book, that has a second part with algebraic topology? I think it is very, very readable. –  Bruno Stonek Apr 6 '12 at 14:12
    
@BrunoStonek I will look for the book. I'm pretty sure I have that book. Thanks –  simplicity Apr 6 '12 at 14:13
    
You might like Lee's book Introduction to Topological Manifolds. –  the symplectic camel Apr 13 '12 at 16:15

2 Answers 2

I think it will help if you "pull the diameter out of the sphere using the 4th dimension" (think about the analogous situation of a diameter in a circle) to see that space is homeomorphic to enter image description here

Now this is similar to the wedge sum of a circle with a sphere which you might have seen before (I think there's a similar example in hatcher). If you want to see the solution to the problem, go here: http://i.imgur.com/afVPm.jpg

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Books on algebraic topology are usually good on giving invariants to show that spaces are not homotopy equivalent, but not so good at showing why spaces are homotopy equivalent. In my book "Topology and groupoids" (2006) there is a chapter on cofibrations, which discusses the homotopy type of adjunction spaces $B \cup_f X$ where $A$ is a closed subspace of $X$ and $f: A \to B$. It is shown that if $f\simeq g$, and $(X,A)$ has the HEP, then $B \cup_f X $ is homotopy equivalent to $B \cup_g X$. You can use this to show your example is homotopy equivalent to $S^2 \vee S^1$, (see p. 293), and one knows the universal cover of this from other examples on this site. There is also a useful gluing theorem for homotopy equivalences, which is quite non trivial to prove, but easy to use.

That book has a different view on covering spaces than other books, by using groupoids systematically. Whether in the time available this can help your exam is another matter! Good luck!

(All these were in the 1968, 1988 editions.)

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