Given a space and a subspace , define . Since the boundary map takes to , it induces a quotient boundary map .
We have a chain complex where holds. The relative homology groups are the homology groups of this chain complex.
Elements of are represented by relative cycles: – chains such that .
A relative cycle is trivial in iff it is a relative boundary: for some and .
Long Exact Sequence (Relative Homology)
There is a long exact sequence of homology groups:
The boundary map is as follows: If a class is represented by a relative cycle , then is the class of the cycle in .