Classify, except isomorphism, all covering spaces with 2 sheets over the space $\mathbb{RP}^2 \times \mathbb{RP}^2$

Question

Classify, except isomorphism, all covering spaces with 2 sheets over the space $\mathbb{RP}^2 \times \mathbb{RP}^2$

I know that $\pi_1(\mathbb{RP}^2 \times \mathbb{RP}^2)=\mathbb{Z}_2 \times \mathbb{Z}_2$ that its abelian.

I have found $p_1: \dfrac{S^2 \times S^2}{c_1} \to \mathbb{RP}^2 \times \mathbb{RP}^2$ and $p_2: \dfrac{S^1 \times S^1}{c_2} \to \mathbb{RP}^2 \times \mathbb{RP}^2$

where $c_1= {(a,1)|a^2=1}$ and $c_2= {(1,b)|b^2=1}$

But I have one missing, since in my homework the teacher said so. Can someone tell me which one? thanks!

Answer 1

Hint: consider $(S^2 \times S^2)/{\sim}$ where $(x_1,x_2) \sim (-x_1,-x_2)$.

I believe this is the orientable double cover of $\mathbb RP^2 \times \mathbb RP^2$ which is nice.

Answer 2

So, resuming what you guys said this would be my proof:

Since $\pi_1(\mathbb{RP}^2\times \mathbb{RP}^2)= \mathbb{Z}_2 \times \mathbb{Z}_2$ its abelian. So $Conj(\mathbb{Z}_2 \times \mathbb{Z}_2)$ are the subgroup of $\mathbb{Z}_2 \times \mathbb{Z}_2$ with index 2

Let $A= <(a,1)|a^2=1>$ and $B=<(1,b)|b^2=1>$

The action of $\mathbb{Z}_2 \times \mathbb{Z}_2$ in $S^2 \times S^2$ its continous and properly discontinous.

Then $\pi : S^2 \times S^2/ \mathbb{Z}_2 \times \mathbb{Z}_2 \cong \mathbb{RP}^2\times \mathbb{RP}^2$ is a covering space and since $S^2 \times S^2$ its simply connected, its the universal covering.

So the covering spaces of two sheetes are:

$p_1: S^2 \times S^2/A \to \mathbb{RP}^2\times S^2 $

$p_2: S^2 \times S^2/B \to S^2 \times \mathbb{RP}^2 $

and let $X : (S^2 \times S^2)/∼$ where $(x_1,x_2) \sim (-x_1,-x_2)$

then $X \to \mathbb{RP}^2\times \mathbb{RP}^2$ such as $[(a,b)] \to ([a],[b])$.

where $[x] =\{ x, -x\}$

but can I say that $S^2 \times \mathbb{RP}^2 \cong \mathbb{RP}^2\times \mathbb{RP}^2$ and $\mathbb{RP}^2 \times S^2 \cong \mathbb{RP}^2\times \mathbb{RP}^2$

because $\mathbb{RP}^2 \times S^2 \to \mathbb{RP}^2\times \mathbb{RP}^2$ with $([a],b) \to ([a],[b])$ and $S^2 \times \mathbb{RP}^2 \to \mathbb{RP}^2\times \mathbb{RP}^2$ with $(a,[b]) \to ([a],[b])$