Contiguous simplicial maps induce the same map on simplicial homology

Let $z$ be an $n$-cycle. Its support generates a finite subcomplex $K_z$ of $K$. Choose a total ordering of the finite vertex set of $K_z$, and use the increasing vertex order as the preferred oriented generator of every simplex of $K_z$. On these generators define $P_k[v_0,\dots ,v_k]:={\sum }_{i=0}^k(-1{)}^i[f(v_0),\dots ,f(v_i),g(v_i),\dots ,g(v_k)],$ omitting a summand when its displayed vertices are not pairwise distinct, and extend linearly. This is a well-defined homomorphism on $C_k(K_z)$ because it is defined on a chosen free basis, and contiguity makes every nondegenerate summand a simplex of $L$.

For each preferred generator of $C_k(K_z)$, expand the boundary of the $i$th prism simplex from step 1.1. Consecutive interior faces cancel, the two outer faces give $g_{\mathrm{\#}}-f_{\mathrm{\#}}$, and the remaining faces give $-P\mathrm{\partial }$. Terms with repeated image vertices cancel in the corresponding normalized formula. Hence $\mathrm{\partial }P+P\mathrm{\partial }=g_{\mathrm{\#}}-f_{\mathrm{\#}}$ on the chains of $K_z$.

Applying step 2.1 to the cycle $z$ gives $g_{\mathrm{\#}}z-f_{\mathrm{\#}}z=\mathrm{\partial }(Pz),$ because $\mathrm{\partial }z=0$. Thus $f_{\mathrm{\#}}z$ and $g_{\mathrm{\#}}z$ represent the same homology class. Every homology class has such a finitely supported cycle representative, so $f_{\ast }=g_{\ast }$ in every degree.