Description and scope of the theory

Up to now, we are only able to derive homology groups for a special kind of space, namely, a compact polyhedron. In chapter 2 we derived homology groups of a simplicial complex and in chapter 3 it was proved that, in a precise and natural sense, the groups are determined, in the case of a finite complex, by the underlying polyhedron. Thus the choice of triangulation of a given polyhedron was revealed as a mere administrative technique designed to associate with the polyhedron a suitable chain complex.

One of the great successes of Combinatorial Topology has been the extension of these techniques to produce homology theories for general topological spaces. A general homology theory may be regarded as satisfactory if it agrees with familiar homology theory on the category of polyhedra and if, moreover, certain central properties of simplicial homology theory hold good also for the general theory.

In this chapter one such homology theory is defined, the singular theory; it is defined by associating with each space a chain complex and then passing to the homology groups of that chain complex (and the contrahomology groups of the adjoint contrachain complex). A continuous map induces homology homomorphisms in an obvious way and we prove that homotopic maps induce the same homomorphisms. To prove this it is convenient to have the so-called cubical singular theory, as well as the simplicial singular theory, and the equivalence of these two theories is established. Singular theory is closely related to homotopy theory; there is a natural homomorphism from homotopy groups to singular homology groups.