Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 2.5.7.13. The Eilenberg-Zilber homomorphism of Example 2.5.7.12 admits a topological interpretation. Recall that, for every simplicial set $S_{\bullet }$, the topological space $| S_{\bullet } |$ is a CW complex (Remark 1.1.8.14). More precisely, $| S_{\bullet } |$ admits a CW decomposition with one cell $e_{\sigma }$ for each nondegenerate simplex $\sigma : \Delta ^ n \rightarrow S_{\bullet }$, where $e_{\sigma }$ is defined as the image of the composite map

\[ | \Delta ^ n |^{\circ } \hookrightarrow | \Delta ^ n | \xrightarrow { | \sigma |} | S_{\bullet } |; \]

here $| \Delta ^ n |^{\circ } = \{ (t_0, \ldots , t_ n) \in \operatorname{\mathbf{R}}_{>0}: t_0 + \cdots + t_ n = 1 \} $ denotes the interior of the topological $n$-simplex. The chain complex $\mathrm{N}_{\ast }( S; \operatorname{\mathbf{Z}})$ of Construction 2.5.5.9 can then be identified with the cellular chain complex associated to this cell decomposition of $| S_{\bullet } |$.

When $S_{\bullet } = S'_{\bullet } \times S''_{\bullet }$ factors a product of two other simplicial sets $S'_{\bullet }$ and $S''_{\bullet }$, the topological space $| S_{\bullet } |$ admits a different CW structure, whose cells are given by $\varphi ^{-1}( e_{\sigma '} \times e_{\sigma ''} )$; here $\varphi $ denotes the canonical map $| S_{\bullet } | \rightarrow | S'_{\bullet } | \times | S''_{\bullet } |$, and $\sigma '$ and $\sigma ''$ range over the collection of nondegenerate simplices of $S'_{\bullet }$ and $S''_{\bullet }$, respectively. The cellular chain complex associated to this cell decomposition can be identified with the tensor product complex $\mathrm{N}_{\ast }( S'_{\bullet }; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }( S''_{\bullet }; \operatorname{\mathbf{Z}})$.

It is not difficult to see that if $\sigma ' \in S'_{p}$ and $\sigma '' \in S''_{q}$ are nondegenerate simplices of $S'_{\bullet }$ and $S''_{\bullet }$, respectively, then the subset $\varphi ^{-1}( e_{\sigma '} \times e_{\sigma ''} ) \subseteq | S_{\bullet } |$ can be written as a finite union of cells of the form $e_{\sigma }$ (where $\sigma $ is a nondegenerate simplex of $S_{\bullet }$). Writing $[\sigma ']$ and $[\sigma '']$ for the corresponding generators of $\mathrm{N}_{p}( S'; \operatorname{\mathbf{Z}})$ and $\mathrm{N}_{q}(S''; \operatorname{\mathbf{Z}})$, the shuffle product is given by

\[ [\sigma '] \triangledown [\sigma ''] = \sum _{\sigma } \pm [ \sigma ] \in \mathrm{N}_{p+q}(S), \]

where the sum is taken over all nondegenerate $(p+q)$-simplices $\sigma $ of $S_{\bullet }$ satisfying $e_{\sigma } \subseteq \varphi ^{-1}( e_{\sigma '} \times e_{\sigma ''})$; note that every such simplex $\sigma $ can be written uniquely as a composition

\[ \Delta ^{p+q} \xrightarrow {\tau } \Delta ^{p} \times \Delta ^{q} \xrightarrow {\sigma ' \times \sigma ''} S'_{\bullet } \times S''_{\bullet } = S_{\bullet } \]

where $\tau $ is a $(p,q)$-shuffle in the sense of Notation 2.5.7.2. Moreover, the sign $(-1)^{\tau }$ also admits a topological interpretation: it is the degree of the open embedding $\varphi |_{ e_{\sigma }}: e_{\sigma } \hookrightarrow e_{\sigma '} \times e_{\sigma ''}$ (with respect to certain standard orientations of the cells $e_{\sigma }$, $e_{\sigma '}$, and $e_{\sigma ''}$).