TITLE: Why Cohomology Groups?
QUESTION [21 upvotes]: Why do we need cohomology groups? Homology groups are easier to compute and given two topological spaces, there is an isomorphism in homology groups if and only if there is an isomorphism in cohomology groups. So why do I need them?

REPLY [7 votes]: Cohomology appears naturaly. Lots of problems are solved locally and then the local solutions glued together to construct global solutions, and cohomology is the precise description of how to do that.