For the pair from part (c), every simplex of dimension below $n$ lies in $L_{n-1}$. Consequently the <relative simplicial chain complex> has
$$
C_k(K_n,L_{n-1})
\cong
\begin{cases}
\mathbb Z,&k=n,\\
0,&k\ne n,
\end{cases}
$$
where the degree-$n$ generator is the class of $[v_0\cdots v_n]$. Its relative boundary is zero because every codimension-one face lies in $L_{n-1}$. All differentials therefore vanish, and
$$
\boxed{
H_k(K_n,L_{n-1})
\cong
\begin{cases}
\mathbb Z,&k=n,\\
0,&k\ne n.
\end{cases}}
$$
This is the <relative homology of a simplex and its boundary>. Equivalently, the <long exact sequence in relative homology>, together with the fact that $|K_n|$ is a <contractible space> and the <homology of a sphere> $|L_{n-1}|\cong S^{n-1}$, gives the <reduced homology> identification $H_k(K_n,L_{n-1})\cong\widetilde H_{k-1}(S^{n-1})$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 20F / d / Solution

{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.

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 Incoming links: Relative homology of a simplex and its boundary