# why is $S^2$ not a Lie group?

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 $S^3$?

• 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. – achille hui May 2 '15 at 17:39

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.

• Hello. You seem to be good at groups. Do you have any hints of any good (freely available) introductions? – mathreadler May 2 '15 at 17:14
• 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. – Duohead Sep 25 '17 at 4:01
• 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) – Max Aug 6 '18 at 6:31