Covering spaces need big help Hatcher 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.


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*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. 
 A: 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 
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 
A: 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.) 
A: Here are a list of books where covers covering spaces:

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*Munkres, James R., Topology., Upper Saddle River, NJ: Prentice Hall. xvi, 537 p. (2000). ZBL0951.54001.


*Lee, John M., Introduction to topological manifolds, Graduate Texts in Mathematics 202. New York, NY: Springer (ISBN 978-1-4419-7939-1/hbk; 978-1-4419-7940-7/ebook). xvii, 433 p. (2011). ZBL1209.57001.


*Lima, Elon Lages, Fundamental groups and covering spaces. Transl. from the Spanish by Jonas Gomes, Natick, MA: A K Peters. ix, 210 p. (2003). ZBL1029.55001.


*Kalajdzievski, Sasho, An illustrated introduction to topology and homotopy, Boca Raton, FL: CRC Press (ISBN 978-1-4398-4815-9/hbk; 978-1-4822-2081-0/ebook). xvi, 469 p. (2015). ZBL1323.55001.


*Ault, Shaun V., Understanding topology. A practical introduction, Baltimore, MD: Johns Hopkins University Press (ISBN 978-1-4214-2407-1/hbk; 978-1-4214-2408-8/ebook). x, 399 p. (2018). ZBL1392.54001.
