Prove that an even dimensional Real Projective space is not orientable.
Help in proving: even dimensional Real Projective space is not orientable
-1
$\begingroup$
differential-geometry
-
3When you post some question, please do not do this in an imperative fashion. Show what was your effort. – 2012-05-01
-
0Here's the answer in 2 dimensions. See if you can generalize to higher dimensions on your own. (The basic arguments are the same). http://math.stackexchange.com/q/133274/22405 – 2012-05-01
-
4You can follow Lee's scketch (Lee's "Introduction to Smooth Manifolds", p. 346-7 - exercises 13-4 and 13-5): first, if $M$ is a connected smoot manifold and $\Gamma$ is a group acting smoothly, freely, and properly on $M$, then $M/\Gamma$ is orientable if and only if every element on $\Gamma$ preserves orientation on $M$. Then, apply this to the case of projevtive spaces, where $M=\mathbb{S}^n$ $\Gamma$ will be $I,-I$ and the quotient will be the projective space itself. – 2012-05-01