Let $K$ be a simplicial complex, and $L$ a subcomplex. As usual, $C_{k}(K)$ denotes the group of $k$-chains of $K$, and $C_{k}(L)$ denotes the group of $k$-chains of $L$.

(a) Let

$$C_{k}(K, L)=C_{k}(K) / C_{k}(L)$$

for each integer $k$. Prove that the boundary map of $K$ descends to give $C_{\bullet}(K, L)$ the structure of a chain complex.

(b) The homology groups of $K$ relative to $L$, denoted by $H_{k}(K, L)$, are defined to be the homology groups of the chain complex $C_{\bullet}(K, L)$. Prove that there is a long exact sequence that relates the homology groups of $K$ relative to $L$ to the homology groups of $K$ and the homology groups of $L$.

(c) Let $D_{n}$ be the closed $n$-dimensional disc, and $S^{n-1}$ be the $(n-1)$-dimensional sphere. Exhibit simplicial complexes $K_{n}$ and subcomplexes $L_{n-1}$ such that $D_{n} \cong\left|K_{n}\right|$ in such a way that $\left|L_{n-1}\right|$ is identified with $S^{n-1}$.

(d) Compute the relative homology groups $H_{k}\left(K_{n}, L_{n-1}\right)$, for all integers $k \geqslant 0$ and $n \geqslant 2$ where $K_{n}$ and $L_{n-1}$ are as in (c).