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.

 

[blue_raver22]

I understand your confusion about covering spaces and the difficulty in determining if a space is a covering space of another. This concept can be tricky to grasp and requires a solid understanding of topology and algebraic topology.

One way to tell if a space is not a covering space of another is by looking at the fundamental group of the spaces. If the fundamental group of the covering space is not isomorphic to the fundamental group of the base space, then it is not a covering space. This is because the fundamental group of a covering space is a subgroup of the fundamental group of the base space.

Another approach is to look at the local properties of the spaces. A covering space must have the property that every point in the base space has a neighborhood that is mapped homeomorphically onto a neighborhood in the covering space. If this property does not hold, then the space is not a covering space.

Furthermore, there are specific criteria that a space must meet in order to be a covering space. For example, the covering map must be a local homeomorphism, meaning that it preserves the local structure of the space. This can be checked by looking at the inverse image of open sets in the base space.

In terms of finding a map that will make a given space a covering space of another, there are specific theorems and techniques that can be used. For example, the lifting criterion states that if a map between two spaces satisfies certain conditions, then it can be lifted to a covering map. Additionally, the Galois correspondence theorem provides a way to construct covering spaces using subgroups of the fundamental group.

Overall, determining if a space is a covering space of another can be a complex process, but with a solid understanding of topology and the relevant theorems, one can confidently identify and construct covering spaces.