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In the Wikipedia page of $SO^+(p,q)$, they do not mention any details about its universal cover. Google searches also did not give any relevant results.

So, what is the universal cover of $SO^+(p,q)$, and how do we prove that it is indeed the universal cover?

Ishan Deo
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  • It's difficult to describe it concretely in general. For low $p$ and $q$, there are some "accidental" descriptions, see here. – Ivo Terek Oct 09 '21 at 16:25
  • Note that $Spin(p,q)$ is a double cover of $SO^+(p,q)$, and thus not the universal cover for $p,q>1$. In the case that $Spin(p,q)$ is the universal cover, it can be explicitly constructed via Clifford algebras. – Kajelad Oct 09 '21 at 17:27
  • @Kajelad Why is $Spin(p,q)$ not the universal cover for $p,q > 1$? – Ishan Deo Oct 09 '21 at 17:32
  • @IshanDeo $Spin(p,q)$ is a double cover of $SO^+(p,q)$, whereas the universal covering is a $|\pi_1(SO^+(p,q))|$-fold covering, where $|\pi_1(SO^+(p,q))|$ is the cardinality of the fundamental group, which is generally greater than 2 (see here). – Kajelad Oct 09 '21 at 21:39

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