Page 76 - Curvature and Homology

5 8 11. TOPOLOGY OF DIFFERENTIABLE MANIFOLDS

Note that a 2-module is simply an abelian group and that for every
integer n
na = a + + a (n times).

Let A be a right A-module and B a left A-module. Denote by
FA,, the free abelian group having as basis the set A x B of pairs
(a, b), a E A, b E B and by r the subgroup of FAX, generated by the
elements of the form
(41 + - (a,,@ - (a2949

The quotient group FAxB/r is known as the tensor product of A and B
and is evidently an abelian group (cf. I.A.4).
There is an operation which may be applied to a p-chain to obtain
a (p - 1)-chain called the boundary operation. It is denoted by 8 and
is defined by the formula

where C, = Xi g, ST and g,[ST : ST-l] is defined by considering G as a
2-module. Moreover, it is linear in Cp(K, G) and hence defines a
homomorphism
a : C,(K,G) -+ C,-,(K,G).
The kernel of a is denoted by Zp(K, G), the elements of which are
called p-cycles. As a consequence of (iv) in the definition of a complex,
a(aC,) = 0 for any C,. The image of C,+,(K, G) under a denoted by
BJK, G) is called the group of bounding p-cycles of K over G and its
elements are called bounding p-cycles or simply boundaries. The quotient
group
H,(K,G) = Z,(K,G)IB,(K,G)
is called the pth homology group of K with coefficient group G. The
elements of H,(K, G) are called homology classes. Clearly, a p-cycle
determines a well-defined homology class. Two cycles I'T and rl in
the same homology class are said to be homologous and we write
rf - I'f. Obviously, Tf - I'f, if and only if, Ff - r: is a boundary.
Assume now that G is the group of integers 2 and write Cp(K) =
Cp(K, Z), etc. The elements of CJK) are called (finite) integral p-chains