How to tell if a space is NOT a covering space of another

[dumbQuestion]

I am just learning about covering spaces and I feel almost every theorem i have to work with starts something like "if you have a covering space p:(\tilde{X}) -> X..." etc. I am a little lost because I'm wondering how I look at a space and then say to myself, what are the possible spaces that cover this? Or alternatively, if I have a space and I'm given another space, I'd like to be able to determine if there's any map p which will make this space a covering space of it.


For example, say I was given S^1 and R. Well, I know R is the universal cover of S^1. But say I didn't realize that. Are there theorems that will let me say confirm that there is some map p that let's R be a covering space of S^1?

Thanks so much!

 

[Bacle2]

There are some results using the induced map on the first fundamental group π₁ of the top and the bottom; if f : X→Y is a covering map, then f_*(π₁)(X,xo) is a subgroup of π₁(Y,yo) , where f(xo)=yo. I can't think of something simpler at the moment. Here f* is the map induced in homotopy , and the fundamental groups are based at the points xo and f(xo) resp .

Also, for univ. covering spaces, the structure of the bottom space must also be in the cover,i.e., the UCover of a manifold is a manifold, same for Lie groups.

In actuality, the result in the 1st paragraph is not so arcane; the fundamental groups of most spaces one runs into in non-specialized applications are known, and then it comes down to some relatively-basic algebra;e.g if the group of the top space is Z --integers-- and that of the bottom is 0 , you know there are no homomorphisms from Z to the 0 group, etc. so, e.g., the circle cannot be a cover for the real line ( tho the opposite is true).
There are also some results with subgroups of the respective groups. For every subgroupof the first fundamental group of the base, there is a covering map; this is a constructive result, in that you can construct the cover,up to , I think, homeomorphism.

Sorry for the drip post; hard to eat while posting. Some definitions:

The induced map on the fundamental group between spaces X and Y takes an element c of the
fundamental class (some curve f: S^1 -->X ) and sends it to the class of f(c). This map is what is called a functor ( the map is well-defined in homotopy, i.e., the image of the map does not depend on the choice of representative of the first homotopy group).

Let me know if you want more defs.

 

[dumbQuestion]

Thanks bacle. The good thing for me is that I am probably more comfortable with the concept of induced homomorphisms then anything else I've come across so far in the last bit of algebraic topology. Thanks a lot for the third paragraph - I feel like this is kind of the logic I'd been using and I feel good that it's been more or less confirmed that this is the correct route to be taking! I was getting really worried I was doing everything wrong and missing something important because I kept focusing on the induced homomorphisms.

 

[Bacle2]

Glad to help.