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.

**Relative cycles**

Elements of are represented by **relative cycles**: – chains such that .

**Relative boundary**

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 .