Page 80 - Curvature and Homology

62 11. TOPOLOGY OF DIFFERENTIABLE MANIFOLDS

that is a complex such that every point of M lies on exactly one simplex
of K and every simplex of K lies on M. This important theorem was
proved by Cairns [17]. The complex K is, of course, not unique. It
follows that M is a polyhedron, that is, M is homeomorphic with I K 1.
Hence, the invariants described above are topological invariants of the
manifold. In the sequel, we shall therefore writte Hp(M, R) for
Hp(K, R), etc.
2.3. Stokes' theorem

Let g, be a singular p-simplex and a a p-form on t5e differentiable
manifold M. Since tp is continuous, the intersection of the carrier of a
and the support df p, is compact. Define the integral of a over
sp = [g, : PO, a**, Pp]
r

For Cp = Xi g, sr, define the integral of a over Cp

by linear extension, that is
C
Jc, la.
Now, let a be a (p - 1)-form over the differentiable manifold M of
dimension n and C, a p-chain of a covering complex K of M. Then,
it can be shown in much the same way as the Stokes' formula was
established in 5 1.6 that

Consider the functional L, defined as follows:

Clearly, La: CJK) -t R is a linear functional, that is La is a p-cochain
with real coefficients. In this way, to a p-form a there corresponds a