As well as the homology groups, one can define cohomology groups . In the common case where each group is isomorphic to for some, we just have, which is again isomorphic to, and, so and determine each other. In general, the relationship between and is only a little more complicated, and is controlled by the universal coefficient theorem. The main advantage of cohomology over homology is that it has a natural ring structure: there is a way to multiply an -dimensional cohomology class by a -dimensional cohomology class to get an -dimensional cohomology class.
Read more about this topic: Homology Theory