Page 81 - Curvature and Homology

p-cochain La. It follows from (2.3.1) that if or is a closed form, La is a
cocycle. Moreover, to an exact form there corresponds a coboundary.
This correspondence between differential forms and cochains may be
extendec! by defining a satisfactory product theory for complexes
(cf. Appendix B).

2.4. De Rham cohomology

Since any two covering complexes of a differentiable manifold M
determine isomorphic homology and cohomology groups we shall call
them the homology and cohomology groups, respectively, of M. Now,
for a fixed closed differential form or of degree p on M the integral
Jr9u is a linear functional on SHp. To see this, put ri = rp + aC,,;
then,

by Stokes' theorem. Hence, there is a unique cohomology class
UP} E HP(M) (= HP(M, R)) such that

for all {rp} E SHp where f P' is a cocyde belonging to the cohomology
class up}. A theorem due to de Rham (cf. Appendix A and [651) implies
that the correspondence or -+ Cfp) establishes an isomorphism (provided
M is compact), that is

(cf. $2.6). Moreover, the cohomology class associated with the exterior
product of two closed differential forms is the cup product of their
cohomology classes (cf. Appendix B). Hence, the isomorphism is a
ring isomorphism. Since the p~ betti number bp(M) of M is the
dimension of the group HP(M), it follows that bp(M) is equal to 'the
number of linearly independent closed dzflerential forms of degree p modulo
the exact forms of depee p. In the remaining sections of this chapter
we shall see how this result was extended by Hodge to a more restricted
class of forms.