= Cellular chain complex of a mapping cone
{title2=$\widetilde C_q(C_f)=C_q(Y)\oplus C_{q-1}(X)$}
For a <cellular map> between nonempty <CW complexes>, the reduced <cellular chain complex> of its <topological mapping cone> has differential
$$
D(y,x)=(d_Yy+f_\#x,-d_Xx).
$$
Here $C_{-1}(X)=0$ and the degree-zero group identifies a vertex $y$ with the reduced chain $y-v$, where $v$ is the cone vertex. Thus this is the <mapping cone> of $f_\#$, in the displayed ordering of the summands.
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