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Say we are given the product spaceX= RP^3 x RP^4 (projective real 3- and 4-space resp.) and we want to find all covering spaces of X. The obvious thing todo would seem to be to use the facts:

i) S^3 covers RP^3; S^4 covers RP^4, both with maps given by the action of Z/2 on

each of the S^n's , i.e., we identify antipodal points in each S^n.

ii) Product of covering spaces is a covering space of the product X

But: how can we guarantee that S^3 XS^4 is the only possible covering space for X?

I think there is a related result dealing with subgroups of the deck/transformation group

( which is Z//2 here ) , but I am not sure.

Any Ideas?

Thanks.

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# Describing All covering Spaces of a Product Space

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