Integration of Forms and de Rham Cohomology

Abstract

In Chapter 6, we studied the first de Rham cohomology H¹(M) of a manifold. This measures the difference between exactness and local exactness of 1-forms on M and was shown to have interesting topological applications. Here we generalize these ideas, using the full Grassmann algebra A*(M) to produce a graded algebra H*(M), the de Rham cohomology algebra. The proper generalization of “locally exact 1-form” is “closed p-form”, defined as a p-form that is annihilated by “exterior differentiation”. Exact forms are closed and H^P(M) measures the extent to which closed p-forms may fail to be exact. By Stokes’ theorem, the geometric boundary operator and exterior differentiation of forms are mutually adjoint operations in a certain precise sense. This is a generalization of the fundamental theorem of calculus and a powerful tool for computing cohomology. The reader who would like to pursue this theory further could hardly do better than to consult [5].