Mapping cone exact sequence ($\cdots \to {\tilde{H}}_{q+1}(C_f;G)\to H_q(X;G)\to H_q(Y;G)\to {\tilde{H}}_q(C_f;G)\to \cdots$)

For nonempty $X$, the topological mapping cone has this long exact sequence in homology, with ordinary homology for $X,Y$ and reduced homology for $C_f$. In degree zero, the augmentation of $X$ is accounted for by the cone vertex. The sequence follows from the Mayer–Vietoris theorem applied to neighborhoods of the cone and $Y$.