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I'm trying to solve the following exercise in Lee's book.

Suppose M is a connected, oriented smooth manifold and Γ is a discrete group acting freely and properly on M. We say the action is orientation-preserving if for each γ ∈ Γ, the diffeomorphism $x\rightarrow γx$ is orientation-preserving. Show that M/Γ is orientable if and only if Γ is orientation-preserving.

Assuming $\Gamma $ to be orientation-preserving, I tried taking a commutative diagram taking an orientation of M to one of M$/\Gamma$ and vice-versa and tried using local diffeomorphism arguments to prove the orientability of M$/\Gamma$. However, I can't make this proof explicit. Nor can I proceed for the other direction.

Can someone please provide a proof for both if and only if parts? Thanks in advance.

adrija
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1 Answers1

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Suppose the quotient space is orientable. Choose an everywhere non-zero top dimensional differential form (section of the determinant bundle)and pull it back by the covering projection. This gives a global top dimensional form on M. Since the covering map is a local diffeomorphism the pull back is also nowhere zero. So M is orientable.

In this case you do not need to assume that the covering transformations are orientation preserving but once you have the global non-zero n-form on M you can show that they must be.

BTW: You can also show this by the naturally of the Stiefel-Whitney classes. If the quotient manifold is orientable then the first Stiefel-Whitney class is zero and so pulls back to zero.

Going the other way. If the group of covering transformations is finite then you can average a global non-zero form over the covering maps and this form will descend to the quotient.It will be everywhere non-zero because the covering transformations preserve orientation.

For the general case, I would try choosing compatible local orientations of the fiber spheres in M ( or in the determinant bundle) over coordinate charts that project diffeomorphically onto charts in the base. Then show that compatible local orientations can be chosen over these projections.

Joe S
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  • Where is the discreteness used in your argument? – Nancium Dec 13 '22 at 02:23
  • Good general answer. Maybe worth pointing out that the naturality of Stiefel-Whitney classes applies to $w_1$ of the pull-back bundle, so unless the pull-back of the tangent bundle is isomorphic to the tangent bundle (which it will be for a finite group action), the conclusion is wrong. For example, any manifold maps to $pt$, which has vanishing $w_1$, but not every manifold is orientable. – Kai Hugtenburg Oct 25 '23 at 09:49