Page 74 - Curvature and Homology

CHAPTER II

TOPOLOGY OF DIFFERENTIABLE MANIFOLDS

In Chapter I we studied the local geometry of a Riemannian manifold
M. In the sequel, we will be interested in how the local properties of M
affect its global behaviour. The Grassman algebra of exterior forms is a
structure defined at each point of a differentiable manifold. In the theory
of multiple integrals we consider rather the Grassman bundle which,
as we have seen, is the union of these algebras taken over the manifold.
It is the purpose of this chapter to describe a class of differential forms
(the harmonic forms) which have important topological implications.
To this end, we describe the topology of M insofar as it is necessary
to define certain algebraic characters, namely the cohomology groups
of M. These groups are, in fact, topological invariants of the
manifold. The procedure followed is similar to that of Chapter I where
the Grassman algebra was first defined over an 'arbitrary' vector space
and then associated with a differentiable manifold via the tangent space
at each point of the manifold. We begin then by defining an abstract
complex K over which an algebraic structure will be defined. We will
then associate K with a related construction K' on M. The cor-
responding algebra over K' will yield the topological invariants we seek.
The chapter is concluded with a theorem relating the class of forms
referred to above with these invariants.

2.1. Complexes

A closure finite abstract ~o&~nplex K is a countable collection of objects
i
{q), = 1, 2, called simplexes satisfying the following properties.
(i) To each simplex Sip there is associated an integer p >= 0 called its
dim-on ;
(ii) To the simplexes Sf and ST-' is associated an integer denoted
by [Sr : 4-7, called their incidence number;
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