Relative homology

In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.

Given a subspace ${\displaystyle A\subseteq X}$, one may form the short exact sequence

where ${\displaystyle C_{\bullet }(X)}$ denotes the singular chains on the space X. The boundary map on ${\displaystyle C_{\bullet }(X)}$ descends^a to ${\displaystyle C_{\bullet }(A)}$ and therefore induces a boundary map ${\displaystyle \partial '_{\bullet }}$ on the quotient. If we denote this quotient by ${\displaystyle C_{n}(X,A):=C_{n}(X)/C_{n}(A)}$, we then have a complex

By definition, the n^th relative homology group of the pair of spaces ${\displaystyle (X,A)}$ is

One says that relative homology is given by the relative cycles, chains whose boundaries are chains on A, modulo the relative boundaries (chains that are homologous to a chain on A, i.e., chains that would be boundaries, modulo A again).

The above short exact sequences specifying the relative chain groups give rise to a chain complex of short exact sequences. An application of the snake lemma then yields a long exact sequence

The connecting map ${\displaystyle \partial }$ takes a relative cycle, representing a homology class in ${\displaystyle H_{n}(X,A)}$, to its boundary (which is a cycle in A).

It follows that ${\displaystyle H_{n}(X,x_{0})}$, where ${\displaystyle x_{0}}$ is a point in X, is the n-th reduced homology group of X. In other words, ${\displaystyle H_{i}(X,x_{0})=H_{i}(X)}$ for all ${\displaystyle i>0}$. When ${\displaystyle i=0}$, ${\displaystyle H_{0}(X,x_{0})}$ is the free module of one rank less than ${\displaystyle H_{0}(X)}$. The connected component containing ${\displaystyle x_{0}}$ becomes trivial in relative homology.