$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Construction (The Alexander-Whitney Construction: Unnormalized Version). Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups with Moore complexes $\mathrm{C}_{\ast }(A)$ and $\mathrm{C}_{\ast }(B)$, respectively. We define a map of graded abelian groups $\overline{\mathrm{AW}}: \mathrm{C}_{\ast }(A \otimes B) \rightarrow \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B)$ by the formula

\[ \overline{\mathrm{AW}}( a \otimes b) = \sum _{0 \leq p \leq n} \iota _{\leq p}^{\ast }(a) \boxtimes \iota _{\geq p}^{\ast }(b) \]

for $a \in A_ n$ and $b \in B_ n$. We will refer to $\overline{\mathrm{AW}}$ as the unnormalized Alexander-Whitney homomorphism.