I am interested in seeing how do cohomology groups of sphere, torus, Mobius band, Klein bootle or projective plane look like? Is there any intuition, like for homology groups?
When it comes to homology, H0 counts number of connected components, H1 counts 1D holes (which is the same as fundamental group, when we talk about path-connected space, and all of those I am interested in actually are), H2 counts 2D holes (how many plugs you need to inflate it). I found here http://www.tricki.org/article/How_to_compute_the_cohomology_of_a_space that when it comes to cohomology, H0 also counts connected components (or path components) and that Hn=$\mathbb{Z}$, when space is connected, compact, orientable n-dimensional manifold.
Can someone list me cohomology groups for sphere, Mobius band, disk, torus, Klein bottle and projective plane and give me some intuition about H1 and H2? I know that Hn=0, for all n>2 for all of them above, because they are 1-dimensional or 2-dimensional manifolds.
Thanks in advance!