Simplicial cone chain contraction
= Simplicial cone chain contraction
{title2=$ds+sd=1$}
In an <augmented chain complex> of a simplex with vertices $v_0,\ldots,v_n$, put $s(1)=[v_0]$ and $s[v_{i_0},\ldots,v_{i_q}]=[v_0,v_{i_0},\ldots,v_{i_q}]$ when $v_0$ is absent, and zero when it is present. Direct expansion of the <boundary operator> gives $ds+sd=1$. This proves that every positive-degree <chain cycle> is a <chain boundary>, without invoking <cellular homology>.