This is from the proof of Theorem 20.7 in "Characteristic classes" by Milnor and Stasheff. Let $f : M^n \rightarrow S^r$ be a smooth map where $M$ is smooth $n$-dimensional manifold and $S^r$ is $r$-dimensional sphere and $n-r = 4i$. The book gives me the diagram commutative up to homotopy.
This is a diagram commutative up to homotopy, where $g$ is piecewise linear and $t, s$ are smooth triangulations and $K$ and $L$ are simplicial complexes. How can we construct the diagram?