Changing the orientation of a connected topological space

[iLoveTopology]

Say we have a disconnected manifold with components C1, C2, C3. (I know in the threat title I said just topological space, but I'm actually thinking of manifolds here, sorry! Not sure how to change the title) It makes intuitive sense that if we're looking at just one of the components, then the orientation has to be the same throughout the component. For example, if there is a closed curve in C1 that is not orientation preserving, then all of C1 is not orientable, which would mean that the entire topological space T is not orientable (by definition: my book says a manifold is non-orientable if there exists some closed curve in it which is not orientation preserving). But say we have a situation where C2 and C3 are orientable. You could cut out the C1 component from from T and now you are left with a space T with components C2 and C3 which are orientable so the entire space is orientable. So in this case we could "cut out" a piece of the manifold (the entire C1 component) and we changed the overall orientation of the space.

But now, if we have a topological space that is just one component, (it is connected), and there is a closed curve in it that is not orientation preserving (so T is not orientable) it doesn't seem like there should be a way to "remove" a piece so that T is now orientable.

Am I off base in this assumption? I am having a hard time finding theorems that will help me prove or disprove this because most the theorems I have in my book have to do with compact spaces and I don't want this to depend on compactness.

 

[quasar987]

Just look at the simplest nonorientable manifold; the Mobius strip. If you cut the strip (i.e. remove a "line"), you are left with a rectangular piece of paper (orientable).

 

[Bacle2]

This may be a bit far out: consider a top form w in your non-orientable manifold. It will be 0 at some point(s), and the set of zeros will be a subspace, since w is a linear map. Look at the kernel of w, and maybe you can understand where orientation fails.

 

[lavinia]

Again with the Moebius band cut it along its equator. Do this with a sissors and see what you get. Is the resulting strip orientable? I would be interested to hear your insights.

After that I would be happy to give you harder examples.

 

[blue_raver22]

I can say that your assumption is correct. In general, the orientability of a topological space is a global property that cannot be changed by removing a subset. This is because the property of orientability is defined in terms of the existence of a consistent orientation for all points in the space.

In the case of a disconnected manifold with components C1, C2, and C3, the orientability of the entire space is determined by the orientability of each individual component. So if we remove the non-orientable component C1, we are left with a space T that is orientable because all of its remaining components are orientable. However, this is a special case and cannot be generalized to all topological spaces.

In the case of a connected topological space, the orientability of the entire space cannot be changed by removing a subset. This is because even if we remove a non-orientable subset, the remaining points in the space may still be non-orientable. In other words, the existence of a non-orientable closed curve in a connected space implies that the entire space is non-orientable, regardless of any subsets that may be removed.

In terms of theorems, the Poincaré-Hopf theorem is a useful tool for determining the orientability of a connected topological space. It states that if a smooth vector field on a compact, connected manifold has a finite number of isolated zeroes, then the Euler characteristic of the manifold is equal to the sum of the indices of these zeroes. This theorem does not depend on compactness, so it can be applied to non-compact spaces as well.

In conclusion, the orientability of a topological space is a global property that cannot be changed by removing a subset. So in the case of a connected topological space, the existence of a non-orientable closed curve implies that the entire space is non-orientable.