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.
Read more about this topic: Singular Homology
Famous quotes containing the words singular, chain and/or complex:
“The singular passion
Abides its object and consumes desire
In the circling shadow of its appetite.”
—Allen Tate (1899–1979)
“To avoid tripping on the chain of the past, you have to pick it up and wind it about you.”
—Mason Cooley (b. 1927)
“All propaganda or popularization involves a putting of the complex into the simple, but such a move is instantly deconstructive. For if the complex can be put into the simple, then it cannot be as complex as it seemed in the first place; and if the simple can be an adequate medium of such complexity, then it cannot after all be as simple as all that.”
—Terry Eagleton (b. 1943)