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I'm reading John Stillwell's "Naive Lie Theory" and it was mentioned there (without giving a proper definition of what a Lie group is) that the only Lie groups among the unit n-spheres are $S^1$ and $S^3$.
Is there a simple or naive explanation of what makes $S^2$ so different from $S^1$ and $S^3$?

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  • $\begingroup$ You should look at answers of this and that, they tell you what sort of topological space is capable to turn into a topological group. $\endgroup$ – achille hui May 2 '15 at 17:39
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Every continuos map $S^2\to S^2$ has at least two fixpoints. But for a group element $\ne 1$, left multiplication has no fixpoints.

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  • $\begingroup$ Hello. You seem to be good at groups. Do you have any hints of any good (freely available) introductions? $\endgroup$ – mathreadler May 2 '15 at 17:14
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    $\begingroup$ Do you mean every continuous map $S^2\to S^2$ homotopic to the identity has two fixed points? The antipodal map is fixed point free. $\endgroup$ – Duohead Sep 25 '17 at 4:01
  • $\begingroup$ Actually there are homeomorphisms of $S^2$ homotopic to the identity with only one fixed point. (E.g. the Mobius transformation given by translation in the complex plane) $\endgroup$ – Max Aug 6 '18 at 6:31

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