Take the tour ×  How I can compute cohomology de Rham of the
projective plane $P^{2}(\mathbb{R})$ using Mayer vietoris or any
other methods?  share|improve this question I retagged your
question because I don't (as of now) see how it relates to
algebraic geometry. –  user641 Mar 28 '11 at 1:12 add comment  1
Answer  up vote 26 down vote accepted  If you remove a point
from $P^2$ you are left with something which looks like a
Moebious band. You can use this to compute $H^\bullet(P^2)$.
Let $p\in P^2$, let $U$ be a small open neighborhood of $p$ in
$P^2$ diffeomorphic to an open disc centered at $p$, and let
$V=P^2\setminus\{p\}$. Now use Mayer-Vietoris.  The cohomology
of $U$ you know. The open set $V$ is diffeomorphic to an open
moebious band, so that tells you the cohomology; alternatively,
you can check that it deformation-retracts to the $P^1\subseteq
P^2$ consiting of all lines orthogonal to the line corresponding
to $p$ (with respect to any inner product in the vector space
$\mathbb R^3$ you used to con