Geometric Interpretation of cochains and cocycles I'm studying cohomology group in A basic course in Algebraic Topology - Massey . My question is : " what is exactly geometric interpretation of cochains and cocycles ? . In his book , he assigns CW-complex as an example but I really don't understand what he have written . Help me figure out , thank you ! 


If you have a simplicial complex (as opposed to a cell complex) you can think of chains as linear combinations of simplices and the boundary operator assigns the sum of $n-1$ dimensional faces to each $n$-simplex, with alternating signs. For cochains, you can also think of them as linear combinations of $n$-simplices, but the coboundary is now the sum of $n+1$-dimensional simplices which have a given $n$ simplex as a face, with certain signs.
Let's take an easy example of a line segment $\tau$ connecting two points $A,B$. Then as a chain $\partial\tau =B-A$, assuming it is oriented from $A$ to $B$, but for cochains we have $\delta B=\tau$ and $\delta A=-\tau$.
I think Massey is probably going through the same construction with cell complexes, but in that case figuring out the coefficients of the boundary operator is a little trickier.
