Page 78 - Curvature and Homology

60 11. TOPOLOGY OF DIFFERENTIABLE MANIFOLDS
The quotient group

is called the pth cohomology group of K with coefficient group G. (It
carries a topology induced by that of D(K, G)). Th? elements of
Hp(K, G) are called cohomology classes. Evidently, a p-cocycle determines
a well-defined cohomology class. Two cocycles fp and gp in the same
cohomology class are said to be cohomologous and we write the
~cohomology' f p -gp. Obviously, f p wgp, if and only if, f p - gp
is a coboundary.

2.2. Singular homology

By a geometric realization KE of an abstract complex K we mean a
complex whose simplexes are points, open line segments, open triangles,
... in an Euclidean space E of sufficiently high dimension corresponding,
respectively, to the 0, 1,2, ***-dimensional objects in K in such a way
that distinct simplexes of K correspond to disjoint simplexes of KE.
The point-set union of all the simplexes of the complex KE written
I KE I is called a polyhedron and the complex K is said to be a covering
of I KE I. Two complexes K and Kt are said to be isomorphic if there is
a 1 - 1 correspondence between their simplexes Sip t-+ Sip preserving
the incidences (cf. definition of an abstract complex). When K and Kt
are isomorphic it can be shown that there is an induced homeomorphism
&: I KE I + I KL I where KE and Ki are geometric realizations of the
complexes K and Kt, respectively such that +Sip = SIP where Sip is
the simplex corresponding to Sip under the isomorphism +. It is indeed
remarkable that the corresponding homology groups of any two covering
complexes of a polyhedron are komorphic. Hence, they are topological
invariants of the polyhedron.
If the coefficients G in the definition of the homology groups form a
ring F, these groups become modules over F. The rank of Hp(K, F)
as a module over F is called the pth betti number bp(K) (= bJK, F))
of the complex K. If F is a field of characteristic zero, these modules are
vector spaces over F. Thus, bJK, F) is the dimension of the vector space
Hp(K, F), that is the maximum number of p-cycles over F linearly
independent of the boundingp-cycles. The expression Z$zoK ( - l)p bp(K)
is called the Euler-Poincart characteristic of K.
Since the homology groups of a covering complex of a polyhedron
are topological invariants of the polyhedron so are the betti numbers
and hence also the Euler-PoincarC characteristic. This, in turn implies