Are the cohomology groups of the real projective plane the same as those of a half of it?

Question

Are the cohomology groups of the real projective plane the same as those if I take the upper hemisphere of it only?

Answer 1

The projective plane does not have "hemispheres." There are at least two standard ways to construct the projective plane from the sphere: (1) as the quotient space of $S^2$ where you identify $(x,y,z) \in S^2$ with $(-x,-y,-z)$, and (2) as the quotient space of the upper hemisphere of points $(x,y,z)$ with $z \geq 0$ where you identify $(x,y,0)$ with $(-x,-y,0)$. Maybe you mean construction (2)? Of course (1) and (2) give exactly the same topological space, the real projective plane, so of course you get the same cohomology either way.