Let $K$ be a simplicial complex with four vertices $v_{1}, \ldots, v_{4}$ with simplices $\left\langle v_{1}, v_{2}, v_{3}\right\rangle$, $\left\langle v_{1}, v_{4}\right\rangle$ and $\left\langle v_{2}, v_{4}\right\rangle$ and their faces.

(a) Draw a picture of $|K|$, labelling the vertices.

(b) Using the definition of homology, calculate $H_{n}(K)$ for all $n$.

(c) Let $L$ be the subcomplex of $K$ consisting of the vertices $v_{1}, v_{2}, v_{4}$ and the 1 simplices $\left\langle v_{1}, v_{2}\right\rangle,\left\langle v_{1}, v_{4}\right\rangle,\left\langle v_{2}, v_{4}\right\rangle$. Let $i: L \rightarrow K$ be the inclusion. Construct a simplicial $\operatorname{map} j: K \rightarrow L$ such that the topological realisation $|j|$ of $j$ is a homotopy inverse to $|i|$. Construct an explicit chain homotopy $h: C_{\bullet}(K) \rightarrow C_{\bullet}(K)$ between $i_{\bullet} \circ j_{\bullet}$ and $\mathrm{id}_{C_{\bullet}(K)}$, and verify that $h$ is a chain homotopy.