Page 33 - Curvature and Homology

(iv) d(dn = 0.
To see this, we need only define

where
u = a(i i,) duil A ... A duit.

In fact, the operator d is uniquely determined by these properties:
Let d* be another operator with the properties (i)-(iv). Since it is linear,
we need only consider its effect upon @ = fduil A ... A dui*. By
property (iii), d*@ = d*f A duil A ... A dui* + fd*(duil A ... A dui9).
Applying (iii) inductively, then (i) followed by (iv) we obtain the desired
conclusion.
It follows easily from property (iv) and (1.4.5) that d(&) = 0 for
any exterior polynomial a of class 2 2.
The operator d is a local operator, that is if a and @ are forms which
coincide on an open subset S of M, then da = dp on S.
The elements A,P(T*) of the kernel of d: AP(T*) -+ AP+l (T*) are
called closed p-forms and the images A,P(T*) of AP-'(T*) under d are
called exact p-forms. They are clearly linear spaces (over R). The
quotient space of the closed forms of degree p by the subspace of exact
p-forms will be denoted by D(M) and called the p-did1 coho-
mology group of M obtained ust'ng dzjbvntial forms. Since the exterior
product defines a multiplication of elements (cohomology classes) in
D(M) and D(M) with values in D+o(M) for all p and q, the direct sum

becomes a ring (over R) called the cohomology ting of M obtained using
differential forms. In fact, from property (iii) we may write

closed form A closed form = closed form,
closed form A exact form = exact form, (1.4.7)
exact form A closed form = exact form.

Examples : Let M be a 3-dimensional manifold and consider the
coordinate neighborhood with the local coordinates x, y, 2. The linear
differential form
u=pcix+qdy+rds (1 A.8)