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Let $H$ be the Hawaiian earring and let H' be the reflection of the Hawaiian earring across the $y$-axis (in the Wikipedia picture). There is a canonical homomorphism from the free product \pi_1(H) * \pi_1(H') to \pi_1(H \cup H') (with basepoint their intersection), but it is not an isomorphism.

This was intended to be a recent homework problem of mine, but as stated the problem actually asked whether the two groups are abstractly isomorphic. I don't know the answer to this question, and neither does my professor. My guess is that they are not isomorphic, but I don't have good intuitions about such large groups.

Edit: to be clear, I know how to do the intended problem, and I also know that H \cup H' is homeomorphic to $H$.

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    @Hans: in this situation there is a distinguished morphism between the two groups. The homework question is whether this morphism is an isomorphism. The question I am asking here is whether _there exists_ an isomorphism.2011-04-17

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The two groups are not isomorphic. See Thm 1.2 of Topology Appl. 123 (2002) 479-505.

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This morphism is not surjective, for the exact same reason $\pi(H)$ is not the free group with countably many generators. Just as there are loops in $\pi_1(H)$ that go through an infinite sequence of circles, here you have loops in \pi_1(H \cup H') that go through an infinite sequence of circles, and who change whose circle they're using an infinite amount of times (for example, just pick the $nth$ circle from the left if $n$ is even, and from the right if $n$ is odd). So for the same reason, the free product doesn't give you all the cycles, and that morphism is not an isomorphism.

However, I'm pretty sure you can find an isomorphism between \pi_1(H \cup H') and $\pi_1(H)$, because they are homeomorphic. For example, send the circle of radius $1/n$ of the $H$ from H \cup H' onto the circle of radius $1/2n$ of $H$ ; and the circle of radius $1/n$ of the H' from H \cup H' onto the circle of radius $1/(2n-1)$ of $H$.

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    Yes, I w$a$s $a$lready aware of both of these facts. The question is whether pi_1(H) is isomorphic to its free square.2011-02-19