TITLE: Covering spaces of $\mathbb{RP}^n\times\mathbb{RP}^n$ for $n > 1$.
QUESTION [5 upvotes]: 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 $ab$?

REPLY [4 votes]: 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).