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Is there a way to prove or understand whether a quotient space of sphere $$S^n/(\mathbf{Z}/k)$$ is a Lie group or not? from some $n$ and $k$.

It seems to me that the only possible Lie groups from the empirical laws telling me that only $S^1=U(1)$, $S^1/(\mathbf{Z}/k)=U(1)$, and $S^3=SU(2)$ and $S^3/(\mathbf{Z}/2)=SO(3)$ can be Lie groups. But I would like to know some general arguments why is that so - or how can $S^n/(\mathbf{Z}/k)$ can be a Lie group?

p.s. After posting the question, I noticed a helpful input from Is there an easy way to show which spheres can be Lie groups? - which we may generalize the statement from there.

Arctic Char
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Марина Marina S
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1 Answers1

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Your question is answered by the following MSE posts:

  • First, if $S^n/(\mathbf{Z}/k)$ has a Lie group structure, then so is $\mathbb S^n$, the universal cover of $S^n/(\mathbf{Z}/k)$ (see here). Thus $n=1$ or $3$ from this.

  • Since $\mathbb S^n \to S^n/(\mathbf{Z}/k)$ is a group homomorphism, $\mathbf Z/k$ is a normal subgroup of $\mathbb S^n$. From this, it must lies in the center. When $n=3$, the center of $SU(2)$ is $\{ \pm I\}$ (see here), so $SU(2)$ and $SO(3)$ are the only options. For $n=1$, you listed already all the options.

Arctic Char
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