# Math Help - Homology and Cohomology

1. ## Homology and Cohomology

What exactly is the difference between homology and cohomology? We know that homology is the number of $n$-cycles that are not $n$-boundaries. This is, essentially the number of holes in a simplicial complex $X$. What is the geometrical interpretation of cohomology?

2. Cohomology classes are exact forms which are not closed. If you integrate a function along a closed surface, the integral depends only on the cohomology class of the integrand. Loosely, I tend to hear that cohomology measures obstructions to creating global solutions from local solutions, say of an equation dx=y. The cohomology class of y measures how this equation fails to have a global solution.

I think that cohomology is less geometric than homology, but the ring structure makes it more useful in many situations.