I've been studying algebraic topology for over half a year now and came across alot of different topics of it (fundamental groups, Van Kampen, singular homology, homology theory, Mayer Vietoris, universal coefficient theorem, knot theory etc.) and recently we started to study cohomolgy in our lecture.
We defined cohomology, proved the universal coefficient theorem and with that we were able to prove quite alot of analogous results which we already proved for homology.
My questions:
Why do we want to study cohomology? Are there some advantages of computing the cohomology of a given topological space compared to computing its homology group? Are there some really suprising/fascinating results which heavily depend on cohomological properties/theorems?
Best Answer
A major issue is the multiplicative structure that is around in cohomology. This allows you to distinguish spaces, which have the same homology. As an example, the $X:=\mathbb CP^2$ and $Y:=S^2\vee S^4$. Then both $X$ and $Y$ are CW-complexes with one cell in dimensions $0$, $2$, and $4$. Hence in both cases the homology with integral coefficients is $\mathbb Z$ in degrees $0$, $2$, and $4$ and $0$ in all other dimensions. This readily implies that also the cohomology groups with integral coefficients are $\mathbb Z$ in degrees $0$, $2$, and $4$ and $0$ in all other dimensions. However, for $X$ the square of a generator of $H^2$ is a generator for $H^4$, whereas for $Y$ this square is zero. (The result for $X$ follows easily, for example, from the fact that the square of the Kähler form on $\mathbb CP^2$ is a volume form, for $Y$ it is kind of obvious that there will be no relation between $H^2$ and $H^4$.)
This shows that $\mathbb CP^2$ is not homotopy equivalent to $S^2\vee S^4$, which in turn implies that the two attaching maps $S^3\to S^2$ used for the two spaces cannot be homotopic, i.e. that the Hopf-fibration is not null-homotopic.
Another issue is that in some situation the fact that cohomology is a contravariant functor is extremely useful. For example, for a topological group $G$ there is a classifying space $BG$ which carries a principal $G$-bundle $EG\to BG$. This has the property that for any sufficently nice space $X$ any principal bundle over $X$ can be written as a pullback $f^*EG$ and $f^*EG\cong g^*EG$ if and only if $f$ and $g$ are homotopic. So a principal bundle gives you a classifying map $f:X\to BG$. Using this map, one can now pull back chomology classes from $BG$ to cohomology classes on $X$. Theses are canonically associated to the bundle, since homotopic maps induce the same pullback in cohomology. This is the topological version of the theory of characteristic classes and it would not work out with homology.