Find all covering spaces of $\mathbb{RP}^n \times \mathbb{RP}^n$, $n>1$

Question

Let $X = \mathbb{RP}^n \times \mathbb{RP}^n$.

I know the following:

• the universal cover of $X$ is $Y = \Bbb S^n \times \Bbb S^n$
• the fundamental group of $X$ is $G = \Bbb Z/2 \Bbb Z \times \Bbb Z/2 \Bbb Z = \{(0,0), (0,1), (1,0), (1,1)\}$
• Covering spaces of $X$ are defined by actions of subgroups of $G$ on $Y$.

Each of the elements of $G$ generates a subgroup of order two. Clearly the covering spaces defined by the action of $\langle(0,1)\rangle$ and $\langle(1,0)\rangle$ on $Y$ are $\Bbb S^2 \times \mathbb{RP}^2$. But what about the action of $\langle(1,1)\rangle$? What covering space does this define? And finally, which of the covering spaces are equivalent? And which are homeomorphic? Thanks.

Answer 1

The nonzero element $1$ of each Cartesian product factor $\Bbb Z / 2 \Bbb Z$ in $G$ acts by the antipodal map, $x \mapsto -x$ on the corresponding sphere. So, by definition, the element $(1, 1) \in G$ corresponds to the space quotient space $$(\Bbb S^2 \times \Bbb S^2) / \!\sim,$$ where $(x, y) \sim (-x, -y)$.

Answer 2

By fundamental theorem of covering spaces, there is a bijective correspondence between conjugacy classes of subgroups of $\pi_1(\Bbb RP^n \times \Bbb RP^n)$ and covers of $\Bbb RP^n \times \Bbb RP^n$.

$\pi_1(\Bbb RP^n \times \Bbb RP^n) \cong \Bbb Z/2 \times \Bbb Z/2$, subgroups of which are precisely the trivial group, the whole group, the two factors isomorphic to $\Bbb Z/2$ and an extra copy of $\Bbb Z/2$ coming from the diagonal group. None of these are conjugate to any other, since $\Bbb Z/2 \times \Bbb Z/2$ is an abelian group.

Thus, any covering space of $\Bbb RP^n \times \Bbb RP^n$ is homeomorphic to either $S^n \times S^n$, $S^n \times \Bbb RP^n$, $S^n \times S^n/(x, y) \sim (-x, -y)$ or itself, and all of these are distinct.