Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 20H (Algebraic Topology)

A simplicial approximation $g:K\to L$ to $f:|K|\to |L|$ is a vertex map satisfying the open-star conditionfor every vertex $v$. Here an open star consists of points whose barycentric coordinate at that vertex is positive. This condition ensures that the images of the vertices of every simplex span a simplex of $L$, and the vertex map extends linearly over simplices.

To triangulate $|K|\times |L|$, order the vertices of each complex consistently. For each product $[v_0,\dots ,v_p]\times [w_0,\dots ,w_q]$, use the simplices whose vertices $(v_i,w_j)$ follow monotone lattice paths from $(0,0)$ to $(p,q)$, taking a step in one coordinate at a time. These prism triangulations agree on faces and cover the product.

If $g,h$ both approximate $f$, choose $x$ in the relative interior of a simplex $\sigma$. Then $x$ belongs to the open star of every vertex $v$ of $\sigma$. Hence $f(x)$ belongs to every open star of $g(v)$ and $h(v)$. Its unique carrier simplex in $L$ contains all these vertices. Thus $g(\sigma )$ and $h(\sigma )$ are faces of that one simplex, proving that $g,h$ are contiguous simplicial maps.

For contiguous simplicial maps, $|g|(x)$ and $|h|(x)$ lie in a common simplex of $L$ for every $x$. Its convexity makesa well-defined map into $|L|$, using the ambient Euclidean realization. The formula is continuous globally and agrees across simplex faces. At $t=0,1$ it equals $|g|,|h|$, so it is the desired homotopy.

Let $F:|K|\times [0,1]\to |L|$ be a homotopy from $f$ to $g$. The inverse images $F^{-1}({\mathrm{st}}_L(w))$ form a finite open cover of the compact product. By the Lebesgue number lemma there is $\delta >0$ such that every subset of diameter less than $\delta$ lies in one of these inverse images.

Choose a barycentric subdivision $K^{(n)}$ with all its open stars of diameter less than $\delta /3$, and times $0=t_0<\cdots <t_m=1$ with gaps less than $\delta /6$. For each vertex $v$ and each time $t_i$, the set(with the evident endpoint conventions) has diameter less than $\delta$ in the product metric. Choose a vertex $w_i(v)$ of $L$ whose open star contains its image under $F$, and set $h_i(v)=w_i(v)$.

At time $t_i$, the star condition makes $h_i$ a simplicial approximation to $F(\cdot ,t_i)$. Adjacent intervals $J_i,J_{i+1}$ overlap, so choose a time in their overlap and a point in the interior of any simplex $\sigma$ of $K^{(n)}$. Its image belongs to the stars of every $h_i(v)$ and $h_{i+1}(v)$ for $v\in \sigma$. As in (i), their vertices lie in one carrier simplex, proving contiguity. Thus the sequence starts with an approximation to $f$, ends with one to $g$, and has contiguous adjacent pairs. Relabeling its indices gives the requested $h_1,\dots ,h_m$.

Here ${\sim }_c$ denotes contiguity, with the endpoint maps approximating $f,g$ respectively.