Topology/Relative Homology

 Topology ← Homology Groups Relative Homology Mayer-Vietoris Sequence →

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

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

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

 Topology ← Homology Groups Relative Homology Mayer-Vietoris Sequence →