Consider the space $\Bbb RP^m \times \Bbb RP^n$, with $m,n>1$. How many classes of connected covering spaces exist for this space? Construct a representant for each class.
What I know: I know the following theorem:
We have that $\pi_1(\Bbb RP^m \times \Bbb RP^n) \cong \Bbb Z_2 \times \Bbb Z_2$. I have some doubts:
Is $p:S^m \times S^n \to \Bbb RP^m \times \Bbb RP^n$ given by $p(x,y)=\{(x,y),(-x,-y)\}$ the universal cover? or is it $p(x,y)=\{(x,y),(-x,-y),(x,-y),(-x,y)\}$?
Are the subgroups of $\Bbb Z_2 \times \Bbb Z_2=\{(0,0),(1,0),(0,1),(1,1)\}$ the trivial subgroup, $\Bbb Z_2 \times \Bbb Z_2$ and $\Bbb Z_2=\{(0,0),(1,0)\}=\{(0,0),(0,1)\}=\{(0,0),(1,1)\}$? If that's right, there are 3 connected covering spaces. How can we compute $(S^m \times S^n)/\Bbb Z_2 \times \Bbb Z_2$ and $(S^m \times S^n)/\Bbb Z_2$?