Page 77 - Curvature and Homology

2.1. COMPLEXES 59
of k. A linear function fP defined on CJK) with values in a com-
mutative topological group G:

is called a p-dimensional cochain or simply a p-cochain. We define groups
dual to the homology groups: The sum of two p-cochains f p and gp is
defined by the formula

for any p-chain C, E C,(K). With this definition of addition the
p-cochains form a group CP(K, G). The inverse of the cochain fp is
the cochain - fP defined by

where - C, is the p-chain (- 1)C,. (This group is actually a topological
group with the following topology: For a p-simplex Sr and an open set
U of G a neighborhood (Sr, U) in O(K, G) is defined as the set of
cochains f p such that f p(Sr) E U). Since the Sip are free generators of
the group CJK), a p-cochain fp defines a unique homomorphism of
CJK) into G.
An operator a* dual to 8 and called the coboundary operator is defined
on the p-cochains as follows:

The image of f P ux?der a* is a (p + 1)-cochain called the coboundary
off p. The operator a* has the properties:
(i) a*(p + gv) = a*p + a*gv,

(ii) a*(a*fP) = 0.
This latter property follows from the corresponding property on chains.
That a* defines a homomorphism

is clear. The kernel of a* is denoted by Zp(K, G) and its elements are
called p-cocycles. The image of 0-l(K, G) under a* denoted by
Bp(K, G) is called the group of coboundingp-cycles or, simply, coboundaries.