Motivations for Homology or Cohomology Theory

Question

In some standard books on Algebraic Topology (except Hatcher's), the motivation for homology or cohomology theory is stated with the help of Cauchy's theorem, Green's theorem, Stoke's theorem etc (for example, one may see in books by Rotman, Massey, Fulton.)

In short, this motivation involves a relationship with "integration".

My question is, is there any other motivation for the study of homology/cohomology?

Answer 1

Sure, lots of them. Here's a scattershot list of some examples, ranging from specific to vague and hand-waving:

• Homology and cohomology are invariant under homotopy equivalence. Hence if two spaces have different homology or cohomology, they're not equivalent. Closed $2$-manifolds, for example, are completely classified by their cohomology.
• We can also talk about the cohomology of groups acting on a module $M$. In addition to the abstract construction, various groups in low-dimensions have particular significance; for example, $H^2(\pi, \mathbb{Z})$ classifies central extensions. By abstract nonsense, $H^*(\pi, \mathbb{Z})$ is just the (say, singular) cohomology of the space $K(\pi, 1)$.
• Similarly, various cohomology theories also appear in algebraic number theory, algebraic geometry, several complex variables, etc.
• In addition to the singular, cellular, etc. cohomology covered in Hatcher, there are also more exotic forms of cohomology. Furthermore, more exotic 'structures', like cobordism, can be constructed as cohomology groups of some complicated space.
• In very fuzzy terms, homology and cohomology measure the extent to which various local solutions of some sort of problem (a differential equation in de Rham cohomology, sections in sheaf cohomology, etc.) can be patched together to provide a global solution. This is the idea (well, an idea, at least) behind obstruction theory. Hatcher doesn't cover it in much detail (it's relegated to one of the appendices of the last chapter), but it's a quite productive area of study.
• We have a natural identification of $[X, K(\pi, n)]$ with $H^n(X, \pi)$, given by $f \to f^*[K(\pi, n)]$. That allows us to convert statements about homotopy into ones about homology, which is generally much easier to compute. As an example of this sort of procedure, real vector bundles over a compact space $X$ are classified by maps $X \to \mathbb{RP}^\infty$, and $\mathbb{RP}^\infty$ happens to be a $K(\mathbb{Z}_2, 1)$. We therefore get a natural bijective correspondence between line bundles over $X$ and $H^1(X, \mathbb{Z}_2)$, and more generally the Stiefel-Whitney class $w(\xi)\in H^*(X, \mathbb{Z}_2)$ of a bundle $\xi \to X$.