Homology Theory - Connection With Integration

Connection With Integration

Suppose that is an open subset of the complex plane, that is a holomorphic function on, and that is a closed curve in . There is then a standard way to define the contour integral, which is a central idea in complex analysis. One formulation of Cauchy's integral theorem is as follows: if and are homologous, then . (Many authors consider only the case where is simply connected, in which case every closed curve is homologous to the empty curve and so .) This means that we can make sense of when is merely a homology class, or in other words an element of . It is also important that in the case where is the derivative of another function, we always have (even when is not homologous to zero).

This is the simplest case of a much more general relationship between homology and integration, which is most efficiently formulated in terms of differential forms and de Rham cohomology. To explain this briefly, suppose that is an open subset of, or more generally, that is a manifold. One can then define objects called -forms on . If is open in, then the 0-forms are just the scalar fields, the 1-forms are the vector fields, the 2-forms are the same as the 1-forms, and the 3-forms are the same as the 0-forms. There is also a kind of differentiation operation called the exterior derivative: if is an -form, then the exterior derivative is an -form denoted by . The standard operators div, grad and curl from vector calculus can be seen as special cases of this. There is a procedure for integrating an -form over an -cycle to get a number . It can be shown that for any -form, and that depends only on the homology class of, provided that . The classical Stokes's Theorem and Divergence Theorem can be seen as special cases of this.

We say that is closed if, and exact if for some . It can be shown that is always zero, so that exact forms are always closed. The de Rham cohomology group is the quotient of the group of closed forms by the subgroup of exact forms. It follows from the above that there is a well-defined pairing given by integration.