Relative homology of a simplex and its boundary
= Relative homology of a simplex and its boundary
{title2=$H_k(\Delta^n,\partial\Delta^n)$}
For $n\geq1$,
$$
H_k(\Delta^n,\partial\Delta^n;\mathbb Z)
\cong
\begin{cases}
\mathbb Z,&k=n,\\
0,&k\ne n.
\end{cases}
$$
In the <relative simplicial chain complex>, every proper face vanishes, leaving one generator in degree $n$ and zero chain groups in all other degrees.