Covering spaces of RP^n x RP^n

[mrbohn1]

Let X = RPⁿ x RPⁿ

I know the following:

- the universal cover of X is Y = Sⁿ x Sⁿ
- the fundamental group of X is G = Z/2Z x Z/2Z = {(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 <(0,1)> and <(1,0)> on Y are S² x RP². But what about the action of <(1,1)>? What covering space does this define?

And finally, which of the covering spaces are equivalent? And which are homeomorphic? Thanks.

 

[pat_connell]

The action of <(1,1)> on Y defines the covering space RP2 x RP2. All of the covering spaces are equivalent, but only RP2 x RP2 and S2 x RP2 are homeomorphic.