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.

Rabson, D. A.; Huesman, J. F.; Fisher, B. N. "Cohomology for Anyone." Found. Phys. 33, 1769-1796, 2003.

CITE THIS AS:

Weisstein, Eric W. "Cohomology." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cohomology.html