Singular cup product on cochains

Let $X$ be a space, let $R$ be a commutative unital ring, and use the positive coboundary $\delta \phi =\phi \mathrm{\partial }$ of Singular cochain complex with coefficients. For $p,q\ge 0$ and cochains $\phi \in C^p(X;R)$, $\psi \in C^q(X;R)$, their cup product is $(\phi ⌣\psi )(\sigma )=\phi (\sigma [0,\dots ,p])\psi (\sigma [p,\dots ,p+q])(\sigma :{\mathrm{\Delta }}^{p+q}\to X)\mathrm{.}$ Extend from simplex generators $R$-linearly. The product is $R$-bilinear in the cochains by distributivity and commutativity in $R$. Zero or negative-degree inputs give zero.

With $J(\phi ,\psi )$ the tensor functional of Additive singular cohomology cross product and $A=\mathrm{AW}$ from Alexander–Whitney map and diagonal approximation, this is exactly $J(\phi ,\psi )A{\mathrm{\Delta }}_{\mathrm{\#}}$. Indeed only the cut of bidegree $(p,q)$ survives. Equivalently, define the AW external cochain by $J(\phi ,\psi )A$ and pull it back along the diagonal. There is no extra cochain sign.

The earlier additive cohomology cross product was expressed through a specified shuffle inverse $T$. The comparison with it is an equality of classes: Alexander--Whitney and shuffle are natural chain-homotopy inverses constructs $K$ with $A-T=dK+Kd$. For cocycles, The additive singular cohomology cross product is well-defined gives $Jd=0$, hence $J(A-T){\mathrm{\Delta }}_{\mathrm{\#}}=JKd{\mathrm{\Delta }}_{\mathrm{\#}}=\delta (JK{\mathrm{\Delta }}_{\mathrm{\#}})\mathrm{.}$ Here postcomposition with the diagonal commutes with boundary, as proved in the diagonal definition. Thus the AW cup class equals diagonal pullback of the earlier external product; equality of the two chosen external cochains is not required. At total degree zero the displayed primitive is zero and the identity is literal. The same tensor-functional identity gives $\delta (JA{\mathrm{\Delta }}_{\mathrm{\#}})=0$ for cocycles, so the compared classes exist.

For degree-zero cochains the formula on a vertex is ordinary multiplication of their values. A constant cochain of value $1$ multiplies any cochain on either side without changing it, including on disconnected spaces. On empty $X$ or over $R=0$ all cochains and products are zero. Face restrictions make sense for every singular simplex, including degenerate ones. A bare abelian coefficient group has no specified multiplication to insert in this formula. No AC or selection of representatives is used in this definition or comparison.