(a) Define the first barycentric subdivision $K^{\prime}$ of a simplicial complex $K$. Hence define the $r^{t h}$ barycentric subdivision $K^{(r)}$. [You do not need to prove that $K^{\prime}$ is a simplicial complex.]

(b) Define the mesh $\mu(K)$ of a simplicial complex $K$. State a result that describes the behaviour of $\mu\left(K^{(r)}\right)$ as $r \rightarrow \infty$.

(c) Define a simplicial approximation to a continuous map of polyhedra

$$f:|K| \rightarrow|L|$$

Prove that, if $g$ is a simplicial approximation to $f$, then the realisation $|g|:|K| \rightarrow|L|$ is homotopic to $f$.

(d) State and prove the simplicial approximation theorem. [You may use the Lebesgue number lemma without proof, as long as you state it clearly.]

(e) Prove that every continuous map of spheres $S^{n} \rightarrow S^{m}$ is homotopic to a constant map when $n<m$.