ABSTRACT
In this chapter, the authors explain the cohomology groups of a simplicial complex. They consider the homomorphism induced by a simplicial map, and the long exact sequence in cohomology. The authors show in some respects, relative cohomology is similar to relative homology; in other respects it is rather different. They deal with some aspects of simplicial cohomology. The topological invariance, indeed the homotopy-type invariance, of the simplicial cohomology groups follows at once. The authors also explain the cohomology groups of some familiar spaces, and can find specific cocycles that generate these groups. The difficulty is that one must go down to the simplicial level and find specific representative cocycles, in order to use the cup product formula. The authors provide an idea of how to picture cochains and cocycles. They define the ring structure of cohomology by giving a specific cochain formula for the multiplication operation.
