Singular Homology - Cohomology

Cohomology

By dualizing the homology chain complex (i.e. applying the functor Hom(-, R), R being any ring) we obtain a cochain complex with coboundary map . The cohomology groups of X are defined as the cohomology groups of this complex; in a quip, "cohomology is the homology of the co- (dual complex)".

The cohomology groups have a richer, or at least more familiar, algebraic structure than the homology groups. Firstly, they form a differential graded algebra as follows:

• the graded set of groups form a graded R-module;
• this can be given the structure of a graded R-algebra using the cup product;
• the Bockstein homomorphism β gives a differential.

There are additional cohomology operations, and the cohomology algebra has addition structure mod p (as before, the mod p cohomology is the cohomology of the mod p cochain complex, not the mod p reduction of the cohomology), notably the Steenrod algebra structure.