Chain boundary
= Chain boundary
{title2=$B_n(C)=\operatorname{im}\partial_{n+1}$}
= Boundary in a chain complex
{synonym}
= Homological boundary
{synonym}
An element of a <chain group> is a boundary when it is the image of a higher-degree element under the <boundary operator>. Since $\partial^2=0$, every boundary is a <chain cycle>. Thus $H_n(C)=Z_n(C)/B_n(C)$, and a <chain cycle> represents zero in <homology (mathematics)> exactly when it is a boundary.