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2.5.8 The Alexander-Whitney Construction

Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups, having normalized Moore complexes $\mathrm{N}_{\ast }(A)$ and $\mathrm{N}_{\ast }(B)$ (Construction 2.5.5.7). In ยง2.5.7, we introduced the Eilenberg-Zilber homomorphism

\[ \mathrm{EZ}: \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \rightarrow \mathrm{N}_{\ast }(A \otimes B) \]

and showed that it induces an isomorphism on homology groups (Theorem 2.5.7.14). The Eilenberg-Zilber homomorphism is usually not an isomorphism of chain complexes. However, it always exhibits the tensor product complex $\mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B)$ as a direct summand of the normalized Moore complex $\mathrm{N}_{\ast }(A \otimes B)$. More precisely, there exist chain maps $\mathrm{AW}: \mathrm{N}_{\ast }(A \otimes B) \rightarrow \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B)$, depending functorially on $A_{\bullet }$ and $B_{\bullet }$, for which the composite map

\[ \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \xrightarrow { \mathrm{EZ} } \mathrm{N}_{\ast }(A \otimes B) \xrightarrow { \mathrm{AW} } \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \]

is equal to the identity. Our goal in this section is to construct these maps and to establish their basic properties.

Notation 2.5.8.1. Let $n$ be a nonnegative integer. For $0 \leq p \leq n$, we define strictly increasing functions

\[ \iota _{\leq p}: [p] \hookrightarrow [n] \quad \quad \iota _{\geq p}: [n-p] \hookrightarrow [n] \]

by the formulae $\iota _{\leq p}(i) = i$ and $\iota _{\geq p}(j) = j+p$. If $A_{\bullet }$ is a simplicial abelian group, we let $\iota _{\leq p}^{\ast }: A_{n} \rightarrow A_{p}$ and $\iota _{\geq p}^{\ast }: A_{n} \rightarrow A_{n-p}$ denote the associated group homomorphisms.

Construction 2.5.8.2 (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.

Proposition 2.5.8.3. Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. Then the unnormalized Alexander-Whitney homomorphism $\overline{\mathrm{AW}}: \mathrm{C}_{\ast }(A \otimes B) \rightarrow \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B)$ is a chain map.

Proof. Let $x$ be an element of the abelian group $\mathrm{C}_{n}(A \otimes B) = A_{n} \otimes B_{n}$; we wish to show that $\partial ( \overline{\mathrm{AW}}(x) ) = \mathrm{AW}^{un}( \partial x)$. Without loss of generality, we may assume that $n>0$ and that $x$ has the form $a \otimes b$, for some elements $a \in A_ n$ and $b \in B_ n$. In this case, we compute

\begin{eqnarray*} \overline{\mathrm{AW}}( \partial (a \otimes b) ) & = & \sum _{i=0}^{n} (-1)^{i} \overline{\mathrm{AW}}( d_ i a \otimes d_ i b ) \\ & = & \sum _{i=0}^{n} \sum _{p=0}^{n-1} (-1)^{i} \iota _{\leq p}^{\ast }( d_ i a) \boxtimes \iota _{\geq p}^{\ast }( d_ i b ) \\ & = & \sum _{i=0}^{n} \sum _{p=0}^{i-1} (-1)^{i} \iota _{\leq p}^{\ast }( d_ i a) \boxtimes \iota _{\geq p}^{\ast }( d_ i b ) + \sum _{i=0}^{n} \sum _{p=i}^{n-1} (-1)^{i} \iota _{\leq p}^{\ast }( d_ i a) \boxtimes \iota _{\geq p}^{\ast }( d_ i b ) \\ & = & \sum _{i=0}^{n} \sum _{p=0}^{i-1} (-1)^{i} \iota _{\leq p}^{\ast }(a) \boxtimes d_{i-p} \iota _{\geq p}^{\ast }(b) + \sum _{i=0}^{n} \sum _{q=i+1}^{n} (-1)^{i} d_{i} \iota _{\leq q}^{\ast }(a) \boxtimes \iota _{\geq q}^{\ast }(b) \\ & = & \sum _{i=0}^{n} \sum _{p=0}^{i} (-1)^{i} \iota _{\leq p}^{\ast }(a) \boxtimes d_{i-p} \iota _{\geq p}^{\ast }(b) + \sum _{i=0}^{n} \sum _{q=i}^{n} (-1)^{i} d_{i} \iota _{\leq q}^{\ast }(a) \boxtimes \iota _{\geq q}^{\ast }(b) \\ & = & \sum _{p=0}^{n} (-1)^{p} \iota _{\leq p}^{\ast }(a) \boxtimes (\sum _{j=0}^{n-p} (-1)^{j} d_ j \iota _{\geq p}^{\ast }(b) ) + \sum _{q=0}^{n} ( \sum _{i=0}^{q} (-1)^{i} d_ i \iota _{\leq q}^{\ast }(a) ) \boxtimes \iota _{\geq q}^{\ast }(b) \\ & = & \sum _{p=0}^{n} (-1)^{p} \iota _{\leq p}^{\ast }(a) \boxtimes \partial \iota _{\geq p}^{\ast }(b) + \sum _{q=0}^{n} \partial \iota _{\leq q}^{\ast }(a) \boxtimes \iota _{\geq q}^{\ast }(b) \\ & = & \partial ( \sum _{p=0}^{n} \iota _{\leq p}^{\ast }(a) \boxtimes \iota _{\geq p}^{\ast }(b) ) \\ & = & \partial ( \overline{\mathrm{AW}}( a \otimes b)). \end{eqnarray*}
$\square$

Proposition 2.5.8.4. The collection of unnormalized Alexander-Whitney homomorphisms $\overline{\mathrm{AW}}: \mathrm{C}_{\ast }(A \otimes B) \rightarrow \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B)$ determine a colax monoidal structure on the Moore complex functor $\mathrm{C}_{\ast }: \operatorname{ Ab }_{\Delta } \rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}})$ (see Variant 2.1.5.11).

Proof. We first show that the unnormalized Alexander-Whitney homomorphisms determine a nonunital colax monoidal structure on the functor $\mathrm{C}_{\ast }$ (Variant 2.1.4.16). By construction, the homomorphism $\overline{\mathrm{AW}}: \mathrm{C}_{\ast }(A \otimes B) \rightarrow \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B)$ is natural in $A_{\bullet }$ and $B_{\bullet }$. It will therefore suffice to show that, for every triple of simplicial abelian groups $A_{\bullet }$, $B_{\bullet }$, and $C_{\bullet }$, the diagram of chain complexes

\[ \xymatrix { \mathrm{C}_{\ast }( A \otimes (B \otimes C) ) \ar [r]^-{\sim } \ar [d]^{ \overline{\mathrm{AW}} } & \mathrm{C}_{\ast }( (A \otimes B) \otimes C) \ar [d]^{\overline{\mathrm{AW}}} \\ \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B \otimes C) \ar [d]^{\operatorname{id}\boxtimes \overline{\mathrm{AW}}} & \mathrm{C}_{\ast }(A \otimes B) \boxtimes \mathrm{C}_{\ast }(C) \ar [d]^{\overline{\mathrm{AW}} \boxtimes \operatorname{id}} \\ \mathrm{C}_{\ast }(A) \boxtimes ( \mathrm{C}_{\ast }(B) \boxtimes \mathrm{C}_{\ast }(C) ) \ar [r]^-{\sim } & (\mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B) ) \boxtimes \mathrm{C}_{\ast }(C) } \]

commutes, where the horizontal maps are given by the associativity constraints of the monoidal categories $\operatorname{ Ab }_{\Delta }$ and $\operatorname{Ch}(\operatorname{\mathbf{Z}})$, respectively. Unwinding the definitions, we see that both the clockwise and counterclockwise composition are given by the construction

\[ a \otimes (b \otimes c) \mapsto \sum _{0 \leq p \leq q \leq n} ( \iota _{\leq p}^{\ast }(a) \boxtimes \rho ^{\ast }(b)) \boxtimes \iota _{\geq q}^{\ast }(c) \]

for $a \in A_ n$, $b \in B_ n$, and $c \in C_ n$, where $\rho $ denotes the nondecreasing map $[q-p] \hookrightarrow [n]$ given by $\rho (i) = i + p$.

Note that the unit object of the category of simplicial abelian groups is the constant functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{ Ab }$ taking the value $\operatorname{\mathbf{Z}}$, which we can identify with the free simplicial abelian group $\operatorname{\mathbf{Z}}[ \Delta ^0]$ generated by the simplicial set $\Delta ^0$. The image of this object under the functor $\overline{\mathrm{AW}}$ is the unnormalized chain complex $\mathrm{C}_{\ast }( \Delta ^0; \operatorname{\mathbf{Z}})$. On the other hand, the unit object of $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ is the chain complex $\operatorname{\mathbf{Z}}[0]$, which we will identify with the normalized chain complex $\mathrm{N}_{\ast }( \Delta ^0; \operatorname{\mathbf{Z}})$. We will complete the proof of Proposition 2.5.8.4 by showing that the quotient map $\epsilon : \mathrm{C}_{\ast }( \Delta ^0; \operatorname{\mathbf{Z}}) \twoheadrightarrow \mathrm{N}_{\ast }( \Delta ^0; \operatorname{\mathbf{Z}})$ is a counit for the nonunital colax monoidal structure constructed above (in the sense of Variant 2.1.5.11). To prove this, we must show that for every simplicial abelian group $A_{\bullet }$, both of the composite maps

\[ \mathrm{C}_{\ast }(A) \simeq \mathrm{C}_{\ast }(A \otimes \operatorname{\mathbf{Z}}[ \Delta ^0] ) \xrightarrow { \overline{\mathrm{AW}} } \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(\Delta ^0; \operatorname{\mathbf{Z}}) \xrightarrow {\operatorname{id}\boxtimes \epsilon } \mathrm{C}_{\ast }(A) \boxtimes \operatorname{\mathbf{Z}}[0] \simeq \mathrm{C}_{\ast }(A) \]

\[ \mathrm{C}_{\ast }(A) \simeq \mathrm{C}_{\ast }(\operatorname{\mathbf{Z}}[ \Delta ^0] \otimes A) \xrightarrow { \overline{\mathrm{AW}} } \mathrm{C}_{\ast }(\Delta ^0;\operatorname{\mathbf{Z}}) \boxtimes \mathrm{C}_{\ast }(A) \xrightarrow {\epsilon \boxtimes \operatorname{id}} \operatorname{\mathbf{Z}}[0] \boxtimes \mathrm{C}_{\ast }(A) \simeq \mathrm{C}_{\ast }(A) \]

are equal to the identity. This follows immediately from the construction (using the fact that $\epsilon $ vanishes on every element of $\mathrm{C}_{\ast }( \Delta ^0; \operatorname{\mathbf{Z}})$ of positive degree). $\square$

We now adapt the Alexander-Whitney construction to the setting of normalized Moore complexes. Recall that, for every simplicial abelian group $A_{\bullet }$, the degenerate simplices of $A_{\bullet }$ generate a subcomplex $\mathrm{D}_{\ast }(A) \subseteq \mathrm{C}_{\ast }(A)$ (Proposition 2.5.5.6) which is a direct summand of $\mathrm{C}_{\ast }(A)$ (Proposition 2.5.6.19). It follows that, if $B_{\bullet }$ is another simplicial abelian group, then we can view $\mathrm{C}_{\ast }(A) \boxtimes \mathrm{D}_{\ast }(B)$ and $\mathrm{D}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B)$ as direct summands of $\mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B)$.

Proposition 2.5.8.5. Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups, and let $K_{\ast } \subseteq \mathrm{C}_{\ast }(A \otimes B)$ be the subcomplex generated by $\mathrm{C}_{\ast }(A) \boxtimes \mathrm{D}_{\ast }(B)$ and $\mathrm{D}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B)$. Then $K_{\ast }$ contains the image of the composite map

\[ \mathrm{D}_{\ast }(A \otimes B) \hookrightarrow \mathrm{C}_{\ast }(A \otimes B) \xrightarrow { \overline{\mathrm{AW}} } \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B). \]

Proof. Let $x$ be an $n$-simplex of the tensor product $A_{\bullet } \otimes B_{\bullet }$, let $0 \leq i \leq n$, and let $s_ i(x)$ denote the associated degenerate $(n+1)$-simplex of $A_{\bullet } \otimes B_{\bullet }$. We wish to show that $\overline{\mathrm{AW}}( s_ i(x) )$ belongs to $K_{\ast }$. Without loss of generality, we may assume that $x = a \otimes b$ for $n$-simplices $a \in A_ n$ and $b \in B_ n$. In this case, we have

\[ \overline{\mathrm{AW}}( s_ i(x) ) = \overline{\mathrm{AW}}( s_ i(a) \otimes s_ i(b) ) = \sum _{p=0}^{n+1} \iota _{\leq p}^{\ast }( s_ i(a) ) \boxtimes \iota _{\geq p}^{\ast }( s_ i(b) ). \]

It will therefore suffice to show that each summand $\iota _{\leq p}^{\ast }( s_ i(a) ) \boxtimes \iota _{\geq p}^{\ast }( s_ i(b) )$ belongs to $K_{\ast }$. This is clear: the simplex $\iota _{\leq p}^{\ast }( s_ i(a) )$ is degenerate if $p > i$, and the simplex $\iota _{\geq p}^{\ast }( s_ i(b) )$ is degenerate for $p \leq i$. $\square$

Construction 2.5.8.6 (The Alexander-Whitney Construction: Normalized Version). Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. It follows from Proposition 2.5.8.5 that there is a unique chain map $\mathrm{AW}: \mathrm{N}_{\ast }(A \otimes B) \rightarrow \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B)$ for which the diagram

\[ \xymatrix { \mathrm{C}_{\ast }(A \otimes B) \ar [r]^-{\overline{\mathrm{AW}}} \ar@ {->>}[d] & \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B) \ar@ {->>}[d] \\ \mathrm{N}_{\ast }(A \otimes B) \ar [r]^-{ \mathrm{AW} } & \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B). } \]

We will refer to $\mathrm{AW}$ as the Alexander-Whitney homomorphism.

We have the following normalized variant of Proposition 2.5.8.4 (which follows immediately from Proposition 2.5.8.4 itself):

Proposition 2.5.8.7. The collection of Alexander-Whitney homomorphisms

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

determine a colax monoidal structure on the normalized Moore complex functor $\mathrm{N}_{\ast }: \operatorname{ Ab }_{\Delta } \rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}})$.

Warning 2.5.8.8. Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. Then we have a canonical isomorphism of simplicial abelian groups $A_{\bullet } \otimes B_{\bullet } \simeq B_{\bullet } \otimes A_{\bullet }$, given degreewise by the construction $a \otimes b \mapsto b \otimes a$. Likewise, there is a canonical isomorphism of chain complexes $\mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \simeq \mathrm{N}_{\ast }(B) \boxtimes \mathrm{N}_{\ast }(A)$ given by the Koszul sign rule (see Warning 2.5.1.14). Beware that these isomorphisms are not compatible with the Alexander-Whitney construction: that is, the diagram

\[ \xymatrix { \mathrm{N}_{\ast }(A \otimes B) \ar [d]^{ \mathrm{AW} } \ar [r] & \mathrm{N}_{\ast }(B \otimes A) \ar [d]^{\mathrm{AW}} \\ \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \ar [r] & \mathrm{N}_{\ast }(B) \boxtimes \mathrm{N}_{\ast }(A) } \]

usually does not commute. Instead, the composite map

\[ \mathrm{N}_{\ast }(A \otimes B) \simeq \mathrm{N}_{\ast }(B \otimes A) \xrightarrow { \mathrm{AW} } \mathrm{N}_{\ast }(B) \boxtimes \mathrm{N}_{\ast }(A) \simeq \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \]

can be identified with the Alexander-Whitney homomorphism associated to the opposite simplicial abelian groups $A_{\bullet }^{\operatorname{op}}$ and $B_{\bullet }^{\operatorname{op}}$. In other words, the colax monoidal structure of Proposition 2.5.8.7 is not a colax symmetric monoidal structure (see Definition ). The same remark applies to the unnormalized Alexander-Whitney construction $\overline{\mathrm{AW}}$ of Construction 2.5.8.2.

Proposition 2.5.8.9. Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. Then the composition

\[ \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \xrightarrow { \mathrm{EZ} } \mathrm{N}_{\ast }(A \otimes B) \xrightarrow { \mathrm{AW} } \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \]

is the identity map.

Proof. Fix element $a \in \mathrm{N}_{p}(A)$ and $b \in \mathrm{N}_{q}(B)$ having shuffle product $a \triangledown b \in \mathrm{N}_{p+q}(A \otimes B)$. We wish to show that the Alexander-Whitney homomorphism $\mathrm{AW}$ satisfies $\mathrm{AW}( a \triangledown b) = a \boxtimes b$. Lift $a$ and $b$ to elements $\overline{a} \in \mathrm{C}_{p}(A) = A_ p$ and $\overline{b} \in \mathrm{C}_ q(B) = B_ q$, respectively. Unwinding the definitions, we see that $\mathrm{AW}( a \triangledown b)$ is given by the image of

\begin{eqnarray*} \overline{\mathrm{AW}}( \overline{a} \bar{\triangledown } \overline{b} ) & = & \overline{\mathrm{AW}}( \sum _{\sigma } (-1)^{\sigma } (\sigma _{-}^{\ast } \overline{a}) \otimes (\sigma _{+}^{\ast }( \overline{b})) ) \\ & = & \sum _{r = 0}^{p+q} \sum _{\sigma } (-1)^{\sigma } (\iota _{\leq r}^{\ast } \sigma _{-}^{\ast })( \overline{a} ) \boxtimes ( \iota _{\geq r}^{\ast } \sigma _{+}^{\ast })(\overline{b}) \end{eqnarray*}

under the quotient map $\mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B) \twoheadrightarrow \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B)$; here the sum is taken over all $(p,q)$-shuffles $\sigma = (\sigma _{-}, \sigma _{+})$ (see Notation 2.5.7.2). Note that the simplex $(\iota _{\leq r}^{\ast } \sigma _{-}^{\ast })(\overline{a}) \in A_{r}$ is degenerate unless $\sigma _{-}(r) = r$ (which implies that $r \leq p$). Similarly, the simplex $( \iota _{\geq r}^{\ast } \sigma _{+}^{\ast })(\overline{b}) \in B_{n-r}$ is degenerate unless $\sigma _{+}(r) = r-p$ (which guarantees that $r \geq p$). We may therefore ignore every term in the sum except for the one with $r = p$ and $\sigma (i) = \begin{cases} (i,0) & \text{ if } i \leq p \\ (p, i-p) & \text{ if } i \geq p, \end{cases}$ for which the corresponding summand is equal to $\overline{a} \boxtimes \overline{b}$ (and therefore has image $a \boxtimes b$ in $\mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B)$). $\square$

Warning 2.5.8.10. Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. Then the unnormalized shuffle product $\bar{\triangledown }$ of Construction 2.5.7.3 induces a chain map $\overline{ \mathrm{EZ} }: \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B) \rightarrow \mathrm{C}_{\ast }(A \otimes B)$. However, the analogue of Proposition 2.5.8.9 for unnormalized Moore complexes is false: that is, the composite map

\[ \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B) \xrightarrow { \overline{ \mathrm{EZ} } } \mathrm{C}_{\ast }(A \otimes B) \xrightarrow { \overline{\mathrm{AW}} } \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B) \]

is usually not equal to the identity.

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. $\square$