Mapping cone exact sequence
= Mapping cone exact sequence
{title2=$\cdots\to\widetilde H_{q+1}(C_f;G)\to H_q(X;G)\to H_q(Y;G)\to\widetilde H_q(C_f;G)\to\cdots$}
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 $Y$.