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$\begingroup$

The probem is:

Suppose $M$ is connected, orientable, smooth manifold and $\Gamma$ is a discrete group acting smoothly, freely, and properly on $M$. We say that the action is orientable-preserving if for each $\gamma \in \Gamma$, the diffeomorphism $x \rightarrow \gamma \cdot x$ is orientation-preserving. Show that $M/ \Gamma $ is orientable if only if the action of $\Gamma$ is orientable-preserving.

Notes: I tried using that $\pi\colon M\to M/\Gamma$ is a covering map. I also tried to use that for any $\gamma$ we have two disjoint open sets $U,V\subset M$ such that $\pi|_{U}$ and $\pi|_{V}$ are diffeomorphisms and $(\pi|^{-1}_{U} \circ \pi|_V)(x)=\gamma.x$ but I didn't prove this result too.

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    Dear Kelson, I noticed that you undid some edits I made yesterday, which is fine, but I just wanted to explain them. (1) Placing the `>` for block quotes in the middle of a math environment results in $\in>$, which I don't think is what you want (2) I think it's different in other languages, but in English I've only ever seen the spelling ["diffeomorphism"](http://en.wikipedia.org/wiki/Diffeomorphism) (3) In specifying the source and target of a morphism it seems to me that `\colon` gives better spacing than `:`. This one is more of a preference, though. (4) "tried use" seems off. Cheers,2012-06-28

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