Cohomology is an invariant of a topological space, formally "dual" to homology, and so it detects "holes" in a space. Cohomology has more algebraic structure than homology, making it into a graded ring (with multiplication given by the so-called "cup product"), whereas homology is just a graded Abelian group invariant of a space. A generalized homology or cohomology theory must satisfy all of the Eilenberg-Steenrod axioms with the exception of the dimension axiom.
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