Let $K$ and $L$ be finite simplicial complexes. Define the $n$-th chain group $C_{n}(K)$ and the boundary homomorphism $d_{n}: C_{n}(K) \rightarrow C_{n-1}(K)$. Prove that $d_{n-1} d_{n}=0$ and define the homology groups of $K$. Explain briefly how a simplicial map $f: K \rightarrow L$ induces a homomorphism $f_{\star}$ of homology groups.

Suppose now that $K$ consists of the proper faces of a 3-dimensional simplex. Calculate from first principles the homology groups of $K$. If a simplicial map $f: K \rightarrow K$ gives a homeomorphism of the underlying polyhedron $|K|$, is the induced homology map $f_{\star}$ necessarily the identity?