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Let $P$ be a real projective plane. Since the fundamental group of $P \times P$ is $Z_2 \times Z_2$ (abelian group with 5 subgroups), there exists five covering spaces.

What is the explicit covering corresponding for $<(1,1)>$? (I found all other 4 cases.)

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up vote 1 down vote accepted

I don't know that this "presentation" is too much more familiar, but the covering space you're interested in is diffeomorphic to $S^2\times S^2/\sim$ where $(x,y) \sim \pm(x,y)$. The projection map sends $[(x,y)]$ to $([x],[y])$.

Now, $S^2\times S^2/\sim$ is diffeomorphic to another relatively well known space: the Grassmanian of unoriented $2$-planes in $\mathbb{R}^4$.

This space is not homotopy equivalent to any of the other covering spaces of $\mathbb{R}P^2\times\mathbb{R}P^2$. Its fundamental group is $\mathbb{Z}/2\mathbb{Z}$, so it could only potentially be homotopy equivalent to $\mathbb{R}P^2\times S^2$ or $S^2\times\mathbb{R}P^2$. On the other hand, both of these second two spaces are nonorientable, while $S^2\times S^2/\sim$ is orientable.

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