Page 83 - Curvature and Homology

Conversely, every such sum represents a value of the integral. The study
of the cochain f becomes a topological problem by virtue of this result,
that is, the problem is to investigate the cycles. As a matter of fact,
homology theory has its origin in this fundamental problem. Another
important property of the cochain f is the following: If a cycle r may be
continuously deformed to a point, then f(r) = 0. This is certainly
the case if D is simply connected.
Now, if r - r', f(r) = f(rt) or, more generally, we may consider
the homology
I' - nl rl + -*- + nz rr, ni E Z (2.5.7)
and it implies that

where the o, are the periods of the form a over the cycles ri. The
values of the line integral are then all of the form f(C) + 2:iF) np,
where bl(D) is the first betti number of D. This is a well-known expres-
sion in analysis. The Cauchy theorem for multiply connected regions
may now be stated: If a is a holomorphic dz#erential and D is a mult+ly
connected region, then
(2.5.9)
for every cycle r - 0 in D.

2.6. Decomposition theorem for compact Riemann surfaces

The following generalizations can be made here. In the first place,
it is possible to consider in place of D a surface with suitably related
integrals. The classical example is the study of abelian integrals

where R(z, w) is a rational function and w = w(z) is an algebraic
function, the integral being evaluated along various paths in the z-plane.
A branch of the function w(z) is chosen at z, and a path from z, to x.
The value of w(z) is then determined by analytic continuation along