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#3: Post edited by user avatar WheatWizard‭ · 2023-11-30T01:20:38Z (12 months ago)
Fixed claim per comment.
  • I'm interested in a proof of the following claim:
  • > If $M$ is an $n$-dimensional compact manifold then the $n$th Betti number, $\beta_n(M) = 1$ if $M$ is orientable and $\beta_n(M) = 0$ otherwise.
  • This claim seems true since it basically says that orientable manifolds have some sort of "inside" while non-orientable manifolds don't. However it seems like a basic enough claim that I would expect to see it somewhere, and I've had trouble finding anywhere making this claim.
  • I did find [this math stackexchange answer](https://math.stackexchange.com/a/4168159/276060) which uses a version of this claim, however the user does not prove it, which would seem to indicate it is elementary.
  • I only understand homology groups at a surface level, so I would appreciate answers that stick closely to first principles. I hope that because this appears elementary that will be possible.
  • I'm interested in a proof of the following claim:
  • > If $M$ is a connected $n$-dimensional compact manifold then the $n$th Betti number, $\beta_n(M) = 1$ if $M$ is orientable and $\beta_n(M) = 0$ otherwise.
  • This claim seems true since it basically says that orientable manifolds have some sort of "inside" while non-orientable manifolds don't. However it seems like a basic enough claim that I would expect to see it somewhere, and I've had trouble finding anywhere making this claim.
  • I did find [this math stackexchange answer](https://math.stackexchange.com/a/4168159/276060) which uses a version of this claim, however the user does not prove it, which would seem to indicate it is elementary.
  • I only understand homology groups at a surface level, so I would appreciate answers that stick closely to first principles. I hope that because this appears elementary that will be possible.
#2: Post edited by user avatar WheatWizard‭ · 2023-11-10T16:24:26Z (about 1 year ago)
  • I'm interested in a proof of the following claim:
  • > If $M$ is an $n$-dimensional manifold then the $n$th Betti number, $\beta_n(M) = 1$ if $M$ is orientable and $\beta_n(M) = 0$ otherwise.
  • This claim seems true since it basically says that orientable manifolds have some sort of "inside" while non-orientable manifolds don't. However it seems like a basic enough claim that I would expect to see it somewhere, and I've had trouble finding anywhere making this claim.
  • I did find [this math stackexchange answer](https://math.stackexchange.com/a/4168159/276060) which uses a version of this claim, however the user does not prove it, which would seem to indicate it is elementary.
  • I only understand homology groups at a surface level, so I would appreciate answers that stick closely to first principles. I hope that because this appears elementary that will be possible.
  • I'm interested in a proof of the following claim:
  • > If $M$ is an $n$-dimensional compact manifold then the $n$th Betti number, $\beta_n(M) = 1$ if $M$ is orientable and $\beta_n(M) = 0$ otherwise.
  • This claim seems true since it basically says that orientable manifolds have some sort of "inside" while non-orientable manifolds don't. However it seems like a basic enough claim that I would expect to see it somewhere, and I've had trouble finding anywhere making this claim.
  • I did find [this math stackexchange answer](https://math.stackexchange.com/a/4168159/276060) which uses a version of this claim, however the user does not prove it, which would seem to indicate it is elementary.
  • I only understand homology groups at a surface level, so I would appreciate answers that stick closely to first principles. I hope that because this appears elementary that will be possible.
#1: Initial revision by user avatar WheatWizard‭ · 2023-11-10T04:52:07Z (about 1 year ago)
Is the nth Betti number determined by orientability?
I'm interested in a proof of the following claim:

> If $M$ is an $n$-dimensional manifold then the $n$th Betti number, $\beta_n(M) = 1$ if $M$ is orientable and $\beta_n(M) = 0$ otherwise.

This claim seems true since it basically says that orientable manifolds have some sort of "inside" while non-orientable manifolds don't. However it seems like a basic enough claim that I would expect to see it somewhere, and I've had trouble finding anywhere making this claim.

I did find [this math stackexchange answer](https://math.stackexchange.com/a/4168159/276060) which uses a version of this claim, however the user does not prove it, which would seem to indicate it is elementary.

I only understand homology groups at a surface level, so I would appreciate answers that stick closely to first principles. I hope that because this appears elementary that will be possible.