Singular Homology

In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups . Intuitively spoken, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.

In brief, singular homology is constructed by taking maps of the standard n-simplex to a topological space, and composing them into formal sums, called singular chains. The boundary operation on a simplex induces a singular chain complex. The singular homology is then the homology of the chain complex. The resulting homology groups are the same for all homotopically equivalent spaces, which is the reason for their study. These constructions can be applied to all topological spaces, and so singular homology can be expressed in terms of category theory, where the homology group becomes a functor from the category of topological spaces to the category of graded abelian groups. These ideas are developed in greater detail below.