Singular Homology - Singular Chain Complex

Singular Chain Complex

The usual construction of singular homology proceeds by defining formal sums of simplices, which may be understood to be elements of a free abelian group, and then showing that we can define a certain group, the homology group of the topological space, involving the boundary operator.

Consider first the set of all possible singular n-simplices on a topological space X. This set may be used as the basis of a free abelian group, so that each is a generator of the group. This set of generators is of course usually infinite, frequently uncountable, as there are many ways of mapping a simplex into a typical topological space. The free abelian group generated by this basis is commonly denoted as . Elements of are called singular n-chains; they are formal sums of singular simplices with integer coefficients. In order for the theory to be placed on a firm foundation, it is commonly required that a chain be a sum of only a finite number of simplices.

The boundary is readily extended to act on singular n-chains. The extension, called the boundary operator, written as

,

is a homomorphism of groups. The boundary operator, together with the, form a chain complex of abelian groups, called the singular complex. It is often denoted as or more simply .

The kernel of the boundary operator is, and is called the group of singular n-cycles. The image of the boundary operator is, and is called the group of singular n-boundaries.

It can also be shown that . The -th homology group of is then defined as the factor group

.

The elements of are called homology classes.