In Hatcher's book Page 79 I was asked to provide two-sheeted covering space $Y \rightarrow X_{1}$ such that the composition $Y \rightarrow X_{1} \rightarrow X$ of the two covering spaces is not a covering space. In here the space $X$ is the shrinking wedge of circles, and $X_{1}$ is placing infinite such spaces onto the line.

(See the figure in the book) http://www.math.cornell.edu/~hatcher/AT/ATpage.html

The example I imagined is this one:

I use $Y$ the same as $X_{1}$, but I make the mapping $Y\rightarrow X_{1}$ like this: I map and the second circle to the first circle, and map the rest to themselves. Then if $Y \rightarrow X_{1} \rightarrow X$ is a covering map, according to the defintion the inverse of a neighborhood in $X$, they must be disjoint; but in here they simply coincide.

I don't know whether this really works as he required. Mostly because the original space is sufficiently bad (not locally path connected) therefore I expect Hatcher would need me to utilize this condition. Also, I want to ask if one can assert that if $X$ is locally path connected, then $Y$ must be a covering space of X. I'm thinking about this because in the next page problem 16, Hatcher asked the reader to prove this:

"16. Give maps $X\rightarrow Y\rightarrow Z$ such that both $Y\rightarrow Z$ and the composition $X\rightarrow Z$ are covering spaces, show that $X\rightarrow Y$ is a covering space if $Z$ is locally path-connected...."

Sorry that this is a purely homework level question.

  • $\begingroup$ Please see the FAQ for a list of sites where you can get help with your homework questions. $\endgroup$ – S. Carnahan Jun 23 '10 at 14:43
  • $\begingroup$ math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Jerzak.pdf $\endgroup$ – Anweshi Jun 27 '10 at 22:24
  • $\begingroup$ Sorry that your question got closed. Though it is an exercise in Hatcher, I believe that it is not trivial to solve it oneself. The pdf linked above solves the question. In any case if you had google searched, you could have found it out yourself. $\endgroup$ – Anweshi Jun 27 '10 at 22:32
  • $\begingroup$ And, a +1. I could not solve it myself when I took the algebraic topology course. So there is at least one more person here who does not think that this question is trivial. $\endgroup$ – Anweshi Jun 27 '10 at 22:35
  • 2
    $\begingroup$ Thanks. I was thinking about getting it deleted, but my browser doesn't work that night, and it remained in here. Being voted down and had topic closed really made myself feel quite awkward. I was thinking maybe that's what it takes to become a graduate student. I am not very interested to find a solution manual for textbook's problems, it makes me feel strange. $\endgroup$ – Kerry Jun 28 '10 at 8:40