The prism operator of a homotopy

Let $H:X\times I\to Y$ be a homotopy from $f$ to $g$ in the sense of Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints. For each singular $n$-simplex $\sigma :{\mathrm{\Delta }}^n\to X$, write $\sigma \times {\mathrm{id}}_I:{\mathrm{\Delta }}^n\times I\to X\times I,(u,t)\mapsto (\sigma (u),t)\mathrm{.}$

For each $0\le i\le n$, let ${\lambda }_i:{\mathrm{\Delta }}^{n+1}\to {\mathrm{\Delta }}^n\times I$ be the affine map sending the vertices of ${\mathrm{\Delta }}^{n+1}$ to $(v_0,0),\dots ,(v_i,0),(v_i,1),(v_{i+1},1),\dots ,(v_n,1)\mathrm{.}$ In barycentric coordinates this is ${\lambda }_i(t_0,\dots ,t_{n+1})=(s_0,\dots ,s_n,t_{i+1}+\cdots +t_{n+1}),$ where $s_k=\{\begin{array}{ll}{t}_k,& k<i,\\ t_i+t_{i+1},& k=i,\\ t_{k+1},& k>i\mathrm{.}\end{array}$

The prism operator determined by $H$ is the degree-$1$ homomorphism $P_H:C_n(X;\mathbb{Z})\to C_{n+1}(Y;\mathbb{Z})$ defined on a singular simplex by $P_H(\sigma ):={\sum }_{i=0}^n(-1{)}^{i}H\circ (\sigma \times {\mathrm{id}}_I)\circ {\lambda }_i,$ and extended linearly. For coefficients in an abelian group $G$, tensor with ${\mathrm{id}}_G$ to obtain $P_H:C_n(X;G)\to C_{n+1}(Y;G)\mathrm{.}$