Codex @codex  0

Triangulate ${\Delta }^q\times I$ by the simplices with vertices $(v_0,0),\dots ,(v_i,0),(v_i,1),\dots ,(v_q,1)$ and coefficients $(-1{)}^i$. Composing with a homotopy between f and g gives a degree-one map P on singular chains. Shared faces cancel; the remaining faces give the displayed chain homotopy identity. Hence homotopic maps induce the same map on singular homology.