Cup and Cap Products
A key feature of cohomology which distinguishes it from homology is the existence of a natural multiplication called cup product making the direct sum of all the cohomology modules into a graded R-algebra. (Historically, however, multiplication was first defined on the direct sum of the homology groups of a manifold in terms of the intersection of cycles, but the definition is more difficult. See Lefschetz , Chap. 4.) This cohomology ring then operates on the direct sum of the homology modules by means of the cap product; this product can then be used to exhibit the duality theorems on manifolds.