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Alexander--Whitney and shuffle are natural chain-homotopy inverses

Statement

For arbitrary spaces $X, Y$ and every commutative unital ring $R$ , the Alexander–Whitney map $A$ and signed shuffle map $S$ are natural chain maps $C_{*} (X \times Y; R) \underset{S}{\overset{A}{⇄}} C_{*} (X; R) \otimes_{R} C_{*} (Y; R) .$ Both preserve degree-zero augmentation, and $A S$ and $S A$ are naturally chain homotopic to the respective identity maps. Moreover, for the specified natural inverse $T$ constructed below there is a natural chain homotopy $A ≃ T$ . This holds for ordinary unnormalized chains and requires no AC.

Facts & Assumptions

[F1]

Alexander–Whitney map and diagonal approximation constructs $A$ , proves its chain-map and naturality identities, and identifies its degree-zero action.

[F2]

The singular chain cross product on generators specifies the finite signed shuffle sum. The singular chain cross product satisfies the boundary formula proves that $S$ is a chain map, and Singular chain cross products are natural proves its naturality.

[F3]

Singular product chain equivalence by simplex models states that $S$ has a natural inverse and natural homotopies, without AC, and that scalar extension preserves the result.

[F4]

The singular chain homotopy formula gives the explicit prism homotopy for a specified homotopy of spaces. Singular cochain complex with coefficients supplies ordinary unnormalized singular chains as free modules on singular simplices.

Proof

Given: Work first over $Z$ . Write $D = C_{*} (X \times Y)$ and $F = C_{*} (X) \otimes C_{*} (Y)$ with the signed tensor differential.

1.1

By [F1] and [F2], $A$ and $S$ are natural chain maps. Both act in degree zero by the inverse identifications between a pair of vertices and their tensor.

F1F2given

1.2

We now construct the model contractions used below. If $Q$ is a standard simplex or a product of two standard simplices and $v$ is its first vertex, the affine homotopy $H (x, t) = (1 - t) v + t x$ stays in $Q$ . Let $p : C (Q) \to C (*)$ collapse $Q$ and let $j : C (*) \to C (Q)$ include $v$ . The prism $P$ of [F4] satisfies $d P + P d = 1 - j p$ . The point complex has one generator $e_{n}$ in each degree, with $d e_{n} = e_{n - 1}$ for positive even $n$ and $d e_{n} = 0$ for odd $n$ . Define $a e_{n} = e_{n + 1}$ for odd $n$ and $a e_{n} = 0$ for even $n$ . Direct substitution gives $d a + a d = 1 - i ϵ$ , where $ϵ : C (*) \to Z [0]$ is augmentation and $i$ includes degree zero. Thus $h_{Q} = P + j a p$ satisfies $d h_{Q} + h_{Q} d = 1 - e_{Q}$ , where $e_{Q} = j i ϵ p$ projects onto the first vertex. For a product model in $F$ , put $h_{F} (x \otimes y) = h_{C} x \otimes y + (- 1)^{∣ x ∣} e_{C} x \otimes h_{E} y .$ The mixed terms cancel, giving $d h_{F} + h_{F} d = 1 - e_{C} \otimes e_{E}$ . Hence in either $D$ - or $F$ -model every positive-degree cycle, and every augmentation-zero degree-zero cycle, has the specified filling $h z$ .

F4givenalgebra

2.1

Construct a natural chain map $T : D \to F$ . In degree zero send the vertex $(x, y)$ to $x \otimes y$ . Suppose $T$ is defined naturally below degree $n > 0$ . On the universal diagonal simplex $a_{n} : Δ^{n} \to Δ^{n} \times Δ^{n}$ , the chain $z = T_{n - 1} d a_{n}$ is a cycle when $n > 1$ and has augmentation zero when $n = 1$ . Define $T_{n} (a_{n}) = h_{F} z$ using step 1.2, and for $σ = (x, y)$ put $T_{n} (σ) = (x_{#} \otimes y_{#}) T_{n} (a_{n})$ . Freeness in [F4] extends this linearly. Then $d T_{n} = T_{n - 1} d$ , and composition of pair maps proves naturality. Thus $T$ is a specified natural chain map whose degree-zero action equals that of $A$ .

F4step 1.1step 1.2

2.2

More generally, let $u, v : E \to G$ be natural chain maps between $D$ or $F$ functors that agree in degree zero. Set $H_{- 1} = 0$ . Assuming $H$ defined below degree $n$ , for each universal degree- $n$ generator $a$ put $z = (u - v) a - H_{n - 1} d a$ . The lower homotopy identity makes $z$ a cycle for $n > 0$ , while at $n = 0$ it is zero. Define $H_{n} (a) = h z$ with the appropriate model contraction from step 1.2, and push forward to arbitrary generators. Then $d H_{n} + H_{n - 1} d = u - v$ ; the specified pushforward rule makes $H$ natural.

F4step 1.2algebra

3.1

Since $T$ and $S$ are natural chain maps and $T_{0} S_{0} = S_{0} T_{0} = 1$ , step 2.2 applied to $(T S, 1_{F})$ and $(S T, 1_{D})$ gives natural homotopies $U$ and $V$ with $d U + U d = T S - 1$ and $d V + V d = S T - 1$ . This independently realizes the existence asserted in [F3].

F2F3step 2.1step 2.2

3.2

The maps $A, T : D \to F$ agree in degree zero by steps 1.1 and 2.1. Applying step 2.2 to $(A, T)$ gives a natural $K$ with $d K + K d = A - T$ . This is the stated comparison with the specified inverse, including its degree-zero normalization.

step 1.1step 2.1step 2.2

4.1

Compose the identity of step 3.2 with the chain map $S$ . Then $d (K S + U) + (K S + U) d = (A - T) S + (T S - 1) = A S - 1,$ and $d (S K + V) + (S K + V) d = S (A - T) + (S T - 1) = S A - 1.$ All summands and composites are natural, so these are the required natural chain homotopies.

F2step 3.1step 3.2

5.1

Tensor these integral identities with $R$ . The canonical identification sends $(x \otimes y) \otimes r$ to $(x \otimes 1) \otimes (y \otimes r)$ ; its inverse sends $(x \otimes a) \otimes (y \otimes b)$ to $(x \otimes y) \otimes a b$ . The tensor relations and commutativity of $R$ make the maps well-defined inverses commuting with the signed differential. The extended maps are the stated AW and shuffle formulas. Additivity of tensoring preserves the homotopy identities without a flatness hypothesis.

F1F2F3step 4.1

6.1

An empty factor or $R = 0$ makes both complexes zero. Degree zero and augmentation were checked in step 1.1; the point calculation in step 1.2 retains the nonzero higher unnormalized chains. Every image is a finite chain because the model contractions are finite prism sums and each recursion uses only finitely many faces and already defined chains. Hence degeneracy, the lowest degree, and coefficient zero impose no exception, and no choice axiom is used.

F3step 1.1step 1.2step 2.1step 2.2step 5.1∎

Depends on

Used by

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Sources

Hatcher §3.2; Miller Lemma 33.2 (standard reference, not scraped)