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The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the first place), there are many more fields in which cohomology is at least a very interesting point of view. Group cohomology is a famous one, and for example it helps in studying extensions.

Here are good points about the "philosophy" behind cohomology. Here are very good, but advanced, ideas on what cohomology "really is".

I would like to ask something a little different:

What are the most unexpected applications of cohomology, or of cohomology-related ideas? Why is cohomology useful/important/interesting when applied to such problems?

Bonus point for real-world applications, or at least outside algebra/geometry/theoretical physics.

Update: Oops, looks like there is a very similar question here, with beautiful answers.

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Did an answer just disappear? – geodude Mar 22 '14 at 17:32

Here is a ridiculous application of cohomology: a proof of $$\sum_{j=0}^n {n \choose j} (-1)^j=0.$$

Let $X=(S_1)^n$ be the $n$-dimensional torus. By the Künneth formula, $H^j(X, \mathbf Q)$ has dimension ${n \choose j}$. Therefore, the Euler characteristic of $X$ is

$$\chi(X)=\sum_{j=0}^n (-1)^j \mathrm{dim}_{\mathbf Q}H^j(X, \mathbf Q) = \sum_{j=0}^n {n \choose j} (-1)^j.$$

On the other hand, $X$ is a compact Lie group; let $\sigma$ be an infinitesimal translation $X \to X$. By the Lefschetz fixed point theorem, $\chi(X)$ is equal to the number of fixed points of $\sigma$, i.e., $0$.

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To me this is an amazing gem. Do you know any other similar applications? Or do you know any ways to convert topological information into elementary number theory information with other tools besides euler characteristic? – Frost Boss Mar 23 '14 at 3:44
@GFrost Thanks! I don't know of any similar reasoning that doesn't lead to something completely trivial in the end. There are many important roles which cohomology plays in number theory, but not quite like this. – Bruno Joyal Mar 23 '14 at 20:42
Of course there are other ways to compute the Euler characteristic of the $n$-dimensional torus, which mirror other ways of proving this identity. For example, the Euler characteristic is multiplicative under products by the Kunneth formula; this corresponds to the proof via $(1 - 1)^n$. – Qiaochu Yuan Nov 7 '14 at 9:07

It all depends on what one means by surprising. This is maybe not so surprising in hindsight, but to me, that the Weil conjectures are proven by étale cohomology is a fantastic application. Cohomology has had a great impact on questions in number theory and surely (different form of cohomologies!) will continue to an important role.

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