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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!

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  • $\begingroup$ See Hatcher chapter 3...he has some nice intuitions. $\endgroup$ Jan 25, 2018 at 16:00

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Cohomology is "basically the same" as homology. To the extent that homology intuitively counts "$n$-dimensional holes", so does cohomology. In fact, the cohomology groups of a space are completely determined by the homology groups, and are often isomorphic to them (though not canonically).

To be more precise, let us assume a space $X$ has finitely generated homology groups in each dimension. Let us write $H_n(X)\cong F_n\oplus T_n$, where $T_n$ is the torsion subgroup of $H_n(X)$ and $F_n$ is a free abelian group. Then the cohomology of $X$ (with coefficients in $\mathbb{Z}$) is given by the formula $$H^n(X)\cong F_n\oplus T_{n-1}.$$ That is, the "free part" of the cohomology is the same as homology, and the "torsion part" is shifted up one dimension from homology. In particular, if the homology is torsion-free, the cohomology is isomorphic to the homology.

(When the homology groups are not finitely generated, or if you use different coefficients, or if you want a natural isomorphism identifying $H^n(X)$, the story is more complicated. The general story is known as the universal coefficient theorem for cohomology.)

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