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I am working on the following problem: for what cardinalities (of $Y$) can $\mathbb{R}^2 - Y$ be a topological group if $Y$ must be finite? My intuition is telling me this somehow involves considering a wedge of circles, but I am not 100% sure on the details. Would anyone be able to confirm this, and if I am wrong, to put me on the right track? Thanks.

YCor
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CBBAM
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Hint: The fundamental group of a topological group is abelian

The question in the body of the question is different from the title though do you want $\mathbf R^3$ or $\mathbf R^2$.

Alex J Best
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    My apologies for the typo, should be $\mathbb{R}^2$. So using the hint, I should try and find the smallest cardinality such that the fundamental group of $\mathbb{R}^2 - Y$ is no longer abelian? – CBBAM Nov 18 '20 at 21:08
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    @CBBAM yes that sounds good! And like you said before its easier to think of retracting to wedges of circles – Alex J Best Nov 18 '20 at 22:35
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    Thank you! So based on that, I am thinking that for $|Y| > 1$ the space cannot be a topological group. My reason is that for any two points, we will obtain a space that is homotopic to the figure 8, which is known to not be abelian. Is something like this correct? Sorry I am still getting the hang of this material. – CBBAM Nov 18 '20 at 23:34
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    @CBBAM yes! That sounds goods to me :) – Alex J Best Nov 19 '20 at 01:24
  • Thank you very much!! – CBBAM Nov 19 '20 at 01:26