ABSTRACT
One of the many ways that relative homology groups can be useful is for giving information about the absolute homology groups. In the early days of algebraic topology, the theorems proved along these lines were often awkward and wordy. The right language for formulating them had not been found. A remarkable algebraic idea due to Eilenberg clarified the matter immensely. Obscure algebraic arguments often become beautifully transparent once they are formulated in terms of exact sequences. Other arguments that were difficult for professional algebraists, become so straightforward that they can be safely left to the reader. The relationship between the absolute and relative homology groups is expressed by an exact sequence called the 'exact homology sequence of a pair'. A long exact sequence is an exact sequence whose index set is the set of integers. That is, it is a sequence that is infinite in both directions. It may, however, begin or end with an infinite string of trivial groups.
