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The fundamental group of $X = \mathbb{RP}^n\times\mathbb{RP}^n$ is just $G=\mathbb{Z}_2\times \mathbb{Z}_2$ when $n > 1$. So connected coverings of $X$ correspond to subgroups of $G$. This has $5$ subgroups: the trivial subgroup, $G$ and $3$ subgroups generated by the elements $a$, $b$ and $ab$.

What is the covering space associated to the subgroup generated by $ab$?

Michael Albanese
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curious
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1 Answers1

4

Think of the diagonal action of $\mathbb{Z}^2$ on $\mathbb{S}^n \times \mathbb{S}^n$ (in other words, acting by antipodal map in both factors).

Igor Rivin
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  • I'd just like to add that, in the case where $n=3$, this cover is homeomorphic to $SO(4)$. In particular, $\mathbb{RP}^3$ is homeomorphic to $SO(3)$, and $SO(4)$ is the appropriate double cover of $SO(3)\times SO(3)$ (see this answer). – Jim Belk Dec 19 '13 at 06:22