Changing the orientation of a connected topological space

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Discussion Overview

The discussion centers on the orientation of connected and disconnected topological spaces, particularly focusing on manifolds. Participants explore the implications of removing components from a manifold on its overall orientability and consider specific examples such as the Möbius strip.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests that within a disconnected manifold, if one component is non-orientable, it can affect the overall orientability of the entire space, but removing that component may result in an orientable space.
  • Another participant points to the Möbius strip as a simple example of a non-orientable manifold, noting that cutting it results in an orientable rectangular piece.
  • A different perspective is introduced regarding a top form in a non-orientable manifold, proposing that examining the kernel of this form may provide insights into where orientation fails.
  • Further discussion on the Möbius band includes a suggestion to cut it along its equator to investigate the orientability of the resulting strip.

Areas of Agreement / Disagreement

Participants express differing views on the implications of removing components from a manifold regarding its orientability. Some agree on the example of the Möbius strip, while others raise questions about the generalizability of these ideas to other non-orientable spaces.

Contextual Notes

The discussion does not resolve the complexities surrounding the definitions of orientability and the conditions under which components can be removed from a manifold. There is an acknowledgment of the limitations of existing theorems, particularly concerning compactness.

Who May Find This Useful

This discussion may be of interest to those studying topology, particularly in the context of manifold theory and orientability, as well as individuals exploring examples of non-orientable spaces.

iLoveTopology
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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.
 
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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).
 
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.
 
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.
 

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