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 $0 \to Z \to Z \to Z/nZ \to 0$. (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 as we talked above
b) from general theory this is a 2-cocyle, (it corresponds to this extension, (it is like "curvature" of connection in differential geometry (can be made precise))).
That is all: we explained why it is group cocycle and what is its role.
I would like to learn this 20 years ago when I learned group cohomology as an undergraduate, but I learned this 1 year 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 of them did not know this also, and after knowing, they share my feeling of surprise.
