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I'm taking a course on Algebraic Topology and I'm struggling to find the solution to this problem:

Let $Y$ be the Hawaiian earring in $\mathbb{R}^2$ and $Y'$ the union on infinite $Y$s moved $3z$ units upward (and downward) with $z \in \mathbb{Z}$ and the line $x=0$ so it is connected. Show a covering $g:Y' \rightarrow Y $. Find a 2-fold covering $f:Y'' \rightarrow Y'$ in a way that $g \circ f: Y'' \rightarrow Y $ is not a covering.

For the first part, I've defined $g$ to send a circumference of radius $1/n$ to the circumference with the radius $1/({n+1})$ in $Y$ and the segments between earrings to the circumference of radius 1. Now, $g$ is a cover, so I have to find the 2-fold cover $f$. I have thought of several covering, but none of them verify that $g \circ f$ is not a cover.

I know that composition of covers is cover if the space is locally simply connected, so the problem must be in the origin of $Y$, where the space is not locally simply connected, but I don't know how to make the composition fail.

Any comment is welcome.

Javier
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1 Answers1

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Hint:

Two rows of Hawaiian earrings, with cross-links Map the cross-links to ever smaller circles, using the fact that you can permute a finite number of circles in the earring homeomorphically.

(If you want the directions to match up, you'll need to cross each set of crosslinks in the middle, but the directions of each circle is not topologically distinguishable anyway, and it would make the figure more complex ...).

hmakholm left over Monica
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  • how did you draw the above? – quantum Jul 06 '18 at 22:57
  • @quantum: As far as I remember, I pilfered a drawing of the Hawaiian earring from Google image search, and edited it in Gimp. – hmakholm left over Monica Jul 07 '18 at 00:25
  • Thanks. It inspired me. So you could use this command in Mathematica (change 10 to 100 or more numbers if you want it realistic): Graphics@Table[Circle[{0, 1/i}, 1/i], {i, 10}] – quantum Jul 10 '18 at 09:36
  • The same answer with a better picture is here: https://math.stackexchange.com/questions/122691/covering-spaces-hatcher-question-6. Note that this is question 1.3.6 from Hatcher's Algebraic Topology. – Cullen Schaffer Jun 01 '19 at 14:03