# Kerodon

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

Corollary 2.5.8.11. Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. Then the Alexander-Whitney homomorphism

$\mathrm{AW}: \mathrm{N}_{\ast }(A \otimes B) \rightarrow \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B)$

is a quasi-isomorphism: that is, it induces an isomorphism on homology.

Proof. By virtue of Proposition 2.5.8.9, the Alexander-Whitney homomorphism is a left inverse to the Eilenberg-Zilber map $\mathrm{EZ}: \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \rightarrow \mathrm{N}_{\ast }(A \otimes B)$, which is a quasi-isomorphism by virtue of Theorem 2.5.7.14. $\square$