# If $S^{2n+1}$ is covering space of $X$, then $X$ is orientable.

Is there any direct way to prove that $n$-manifold is orientable? In AT we can just calculate $n$'th homology group and check whether it's $\mathbb Z$ or $0$. But I want a geometric method, using differential forms. Thanks!

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Are you intending to do any of your homework yourself or have you posted it all? – Alex B. Jun 24 '11 at 10:38
Sorry,I have thought these problems for a long time,but I can't find any idea, so I just have to find help... – henry Jun 25 '11 at 12:20
Interesting question, btw (one in the title, I mean). – Grigory M Jun 26 '11 at 12:11

@henry "so degree is 1" -- why not $-1$? (And are you using that sphere is odd, not even dimensional?) – Grigory M Jun 29 '11 at 20:33