# Describing All covering Spaces of a Product Space

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1. Aug 26, 2011

### Bacle

Hi, All:

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.

2. Aug 27, 2011

### mathwonk

covering spaces are classified by the fundamental group, so you want to know the fundamental group of the product space. that is fairly easy.

3. Aug 27, 2011

### Bacle

Thanks; but what do I do after I calculate Pi_1 ? I know there is some result with

its subgroups, but I am not clear on what that is. Any refs/ name of result, please?

4. Aug 31, 2011

### mathwonk

classification of covering spaces by fundamental group ???

5. Sep 2, 2011

### mathwonk

if you need more help just ask.

6. Sep 2, 2011

### Bacle

That's O.K. Mathwonk, thanks; from what I got, we mod out the universal covering spaces
by all subgroups of the fundamental group ; in our case, we have the product