Homology groups are examples of topological invariants: topologically equivalent spaces have the same homology groups. The idea behind homology groups is to consider a special family of topological spaces C for which the concept of a boundary makes sense, namely spaces made of simple polyhedra, and to use maps from those spaces into another topological space X as a way of exploring the topology of X. Roughly, the nth homology group of X describes continuous maps into X from those special n-dimensional spaces C that cannot be extended to a continuous map into X from any of the special (n+1)-dimensional spaces whose boundary is C. This article introduces homology groups. A brief overview of related topological invariants called cohomology groups and cohomology rings is also included.
