# Topology/Relative Homology

< Topology

Let the notation represent the singular chains for X then, for a subspace there exists a short exact sequence

meaning we can define the *relative homology* as .

< Topology

Topology | ||

← Homology Groups | Relative Homology |
Mayer-Vietoris Sequence → |

Let the notation $C_{\bullet }(X)$ represent the singular chains for X then, for a subspace $A\subset X$ there exists a short exact sequence

- $0\to C_{\bullet }(A)\to C_{\bullet }(X)\to C_{\bullet }(X)/C_{\bullet }(A)\to 0$

meaning we can define the *relative homology* as $H_{n}(X/A)\cong H_{n}(C_{\bullet }(X)/C_{\bullet }(A))$.

Topology | ||

← Homology Groups | Relative Homology |
Mayer-Vietoris Sequence → |