The singular chain cross product on generators

Let $\sigma :{\mathrm{\Delta }}^p\to X$ and $\tau :{\mathrm{\Delta }}^q\to Y$ be singular simplices. A $(p,q)$-shuffle is a permutation $\theta$ of $\{1,\dots ,p+q\}$ such that $\theta (1)<\cdots <\theta (p)\text{and}\theta (p+1)<\cdots <\theta (p+q)\mathrm{.}$ Each shuffle determines a monotone lattice path from $(0,0)$ to $(p,q)$, hence an affine simplex ${\lambda }_{\theta }:{\mathrm{\Delta }}^{p+q}\to {\mathrm{\Delta }}^p\times {\mathrm{\Delta }}^q$ whose vertices are the successive vertices of that path in the product simplex.

The singular chain cross product on generators is $\sigma \times \tau :={\sum }_{\theta \in \mathrm{Sh}(p,q)}\mathrm{sgn}(\theta )(\sigma \times \tau )\circ {\lambda }_{\theta }\in C_{p+q}(X\times Y;\mathbb{Z}),$ where $\sigma \times \tau :{\mathrm{\Delta }}^p\times {\mathrm{\Delta }}^q\to X\times Y$ is the product map.

Extend bilinearly to $\times :C_p(X;\mathbb{Z})\times C_q(Y;\mathbb{Z})\to C_{p+q}(X\times Y;\mathbb{Z})\mathrm{.}$ For $p=0$ or $q=0$, this agrees with the evident product of a point simplex and an arbitrary simplex.