A <simplicial map> $g:K\to L$ is a <simplicial approximation> to $f:|K|\to|L|$ when
$$
f(\operatorname{st}_K(v))
\subseteq\operatorname{st}_L(g(v))
$$
for every vertex $v$, where $\operatorname{st}$ denotes the <open star in a simplicial complex>.

Fix $x$ in a simplex with vertices $v_0,\ldots,v_q$, retaining only vertices with positive barycentric coordinate at $x$. Then $x\in\operatorname{st}_K(v_i)$ for every $i$, so $f(x)$ lies in every $\operatorname{st}_L(g(v_i))$. The vertices $g(v_i)$ and the vertices carrying the barycentric coordinates of $f(x)$ consequently span a common simplex of $L$. Both $f(x)$ and $|g|(x)$ lie in its convex realization. The <straight-line homotopy from a simplicial approximation>
$$
\boxed{H(x,t)=(1-t)f(x)+t|g|(x)}
$$
is therefore well-defined and continuous, with endpoints $f$ and $|g|$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 21H / c / Solution

If $g$ simplicially approximates $f$, then $f(x)$ and $|g|(x)$ lie in a common simplex for every $x$. The affine formula
$$
H(x,t)=(1-t)f(x)+t|g|(x)
$$
therefore remains in the realization and gives a homotopy from $f$ to $|g|$.


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 Incoming links: Straight-line homotopy from a simplicial approximation