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Which of the projective spaces

$$\Bbb R\Bbb P^n, \quad \Bbb C\Bbb P^n,\quad\Bbb H\Bbb P^n$$

admits the structure of a topological group/Lie group (compatible with its usual topology)?

Trivially, $\Bbb R\Bbb P^2\cong\Bbb S^1$ does, as it can be interpreted as the unit complex numbers. According to this answer, the $\Bbb C\Bbb P^n$ do not admit a Lie group structure for any $n\ge 2$. What about a topological group structure?

What about the others?

Matt Samuel
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M. Winter
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  • It should be the case that if it has a topological group structure, then it has a Lie group structure according to this answer here. – jgon Nov 04 '19 at 17:25
  • Some necessary conditions to admit a Lie group structure can be found in the answer here. – jgon Nov 04 '19 at 17:27
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    Jason DeVito's comment on the answer I linked to says that a compact Lie group is either abelian (and thus a torus), or nonabelian and have nonzero $H^3$. Thus $\Bbb{H}P^n$ and $\Bbb{C}P^n$ are not Lie groups unless they are torii. – jgon Nov 04 '19 at 17:32
  • The tangent bundle of a Lie group is necessarily trivial, which allows you to exclude many spaces from admitting Lie groups structures. – Thomas Rot Nov 05 '19 at 08:26

1 Answers1

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A topological group structure lifts to a covering space. Therefore, other than $\mathbb RP^1$ the only real projective space with a group structure is $\mathbb RP^3\simeq SO(3)$.

Matt Samuel
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  • Thank you very much! Since this is far from my expertise, what would be the easiest source to cite for this claim? I suspect that the first sentence is just common knowledge in this area, so any standard book shall do. What about the second part? Do I get this right: by a topological group structure on any of these spaces, we also get one on a certain sphere, and the spheres with such a structure are classified. – M. Winter Nov 04 '19 at 17:39
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    @M.W That's right. I don't know a text of the top of my head, but there's actually an MSE question about it from 9 years ago. Hard to link on mobile, it's called "Covering of a topological group is a topological group" – Matt Samuel Nov 04 '19 at 17:42
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    Actually, $\mathbb R P^3$ is diffeomorphic to $SO(3)$, not $SU(2)$. It’s double covered by $SU(2)$. – Jack Lee Nov 04 '19 at 19:33
  • @Jack Thanks, corrected. – Matt Samuel Nov 04 '19 at 19:34