= Mapping cone acyclicity criterion
{title2=$H_*(M(f))=0\iff f\text{ is a quasi-isomorphism}$}
For a <chain map> $f:C\to D$, use $M_i=C_{i-1}\oplus D_i$ and $d(x,y)=(d_Cx,(-1)^if(x)+d_Dy)$. The <chain map> identity makes $d^2=0$. The <short exact sequence of chain complexes> $0\to D\to M\to C_{*-1}\to0$ has <connecting homomorphism> $(-1)^if_*$ in degree $i$. Its <long exact sequence in homology> shows that the <homology> of $M$ vanishes precisely when every $f_*$ is an <isomorphism>.
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