Occurrences of (co)homology in other disciplines and/or nature
Actually even schoolchildren calculate group co-cycle. (Without knowing that it is called like this). Cohomology occurs in everyday life as soon as one learns to count.
5+7 = 1 2
4 + 5 = 0 9
2 + 8 = 1 0
What is the function on which sends a pair (a,b) to the $0$ or $1$ depending result is greater than 9 or not ? ( e.g. f(5,7)= 1, f(4,5) = 0, f(2,8)= 1).
This is actually a 2-cocycle for group $Z/nZ$ with values in $Z$.
It can be checked directly or...
Let us look on it more conceptually. Consider the standard short exact sequence of abelian groups $0 \to Z \to Z \to Z/n \to 0$. (First map is multiplication by $n$, the second is factorization and will be denoted by $p$).
Choose section $s: Z/nZ \to Z$ (i.e. any map such $ps=Id$, where $p: Z \to Z/nZ$, it is like connection in differential geometry (can be made precise)).
Define $f(a,b)= s(a)+s(b) - s(a+b)$
Note that: a) this function $f(a,b)$ is exactly we talked above
b) from general theory this is 2-cocyle, (it corresponds to this extension, (it it is like "curvature" of connection is differential geomety (can be made precise)).
That is all: we explained why it is group cocycle and what its role.
I would like to learn this 20 years ago when I learned group cohomology as undergraduate, but I learned this 1 ago, doing some engineering work in wireless communication... I am still surprised that it is not written on the first page of any textbook which deals with group cohomology, when I am explaining this to my friends most did not know this also and after knowing share my feeling of surprise.
Robert Ghrist is all about applied topology: Sensor Network, Signal Processing, and Fluid Dynamics. (homepage: http://www.math.upenn.edu/~ghrist/index.html ). For instance, we want to use the least number of sensors to cover a certain area, such that when we remove one sensor, a part of that area is undetectable. We can form a complex of these sensors and hence its nerve, and use homology to determine whether there are any gaps in the sensor-collection. I've met with him in person and he expressed confidence that this is going to be a big thing of the future.
There are also applications of cohomology to Crystallography (see Howard Hiller) and Quasicrystals in physics (see Benji Fisher and David Rabson). In particular, it uses cohomology in connection with Fourier space to reformulate the language of quasicrystals/physics in terms of cohomology... Extinctions in x-ray diffraction patterns and degeneracy of electronic levels are interpreted as physical manifestations of nonzero homology classes.
Another application is on fermion lattices (http://arxiv.org/abs/0804.0174v2), using homology combinatorially. We want to see how fermions can align themselves in a lattice, noting that by the Pauli Exclusion principle we cannot put a bunch of fermions next to each other. Homology is defined on the patterns of fermion-distributions.
Recently, it was realized that quantum many-body states can be divided into short-range entangled states and long-range entangled states.
The quantum phases with long-range entanglements correspond to topologically ordered phases, which, in two spatial dimensions, can be described by tensor category theory (see cond-mat/0404617). Topological order in higher dimensions may need higher category to describe them.
One can also show that the quantum phases with short-range entanglements and symmetry $G$ in any dimensions can be "classified" by Borel group cohomology theory of the symmetry group. (Those phases are called symmetry protected trivial (SPT) phases.)
The quantum phases with short-range entanglements that break the symmetry are the familar Landau symmetry breaking states, which can be described by group theory.
So, to understand the symmetry breaking states, physicists have been forced to learn group theory. It looks like to understand patterns of many-body entanglements that correspond to topological order and SPT order, physicists will be forced to learn tensor category theory and group cohomology theory. In modern quantum many-body physics and in modern condensed matter physics, tensor category theory and group cohomology theory will be as useful as group theory. The days when physics students need to learn tensor category theory and group cohomology theory are coming, may be soon.