LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Cartan coherence for higher diagonals

Statement

Work over $F_{2}$ . Put $C = C_{*} (X)$ , $E = C_{*} (Y)$ , and let $A : C_{*} (X \times Y) \to C \otimes E$ and $S : C \otimes E \to C_{*} (X \times Y)$ be Alexander--Whitney and shuffle. If $T$ interchanges the two factors output by a higher diagonal and $τ$ regroups

(C \otimes C) \otimes (E \otimes E) ⟶ (C \otimes E) \otimes (C \otimes E),

define degree- $i$ maps

L_{i} = (A \otimes A) D_{i}^{X \times Y} S, R_{i} = \sum_{r + s = i} τ (D_{r}^{X} \otimes T^{r} D_{s}^{Y}) .

Let $Q$ interchange the two $C \otimes E$ blocks. There are natural degree- $(i + 1)$ maps $H_{i}$ , with $H_{- 1} = 0$ , such that

L_{i} - R_{i} = d H_{i} + H_{i} d + (1 + Q) H_{i - 1} .

These homotopies preserve both relative carriers: for $B \subseteq X$ and $D \subseteq Y$ , they carry $C_{*} (B) \otimes E$ into $(C_{*} (B) \otimes E)^{\otimes 2}$ and $C \otimes C_{*} (D)$ into $(C \otimes C_{*} (D))^{\otimes 2}$ .

Facts & Assumptions

Given: Spaces $X, Y$ , their ordinary unnormalized mod-two singular chains, and a fixed natural higher-diagonal system.

[F1]

The higher diagonals satisfy $d D_{i} + D_{i} d = (1 + T) D_{i - 1}$ (Natural higher diagonal approximations).

[F2]

The Alexander--Whitney formula is a finite sum of tensor products of face restrictions (Alexander–Whitney map and diagonal approximation).

[F3]

The shuffle $S$ and Alexander--Whitney $A$ are natural chain-homotopy inverses on ordinary unnormalized chains without AC (Alexander--Whitney and shuffle are natural chain-homotopy inverses).

[F4]

A specified homotopy has a finite prism operator satisfying the singular chain-homotopy identity (The singular chain homotopy formula).

[F5]

The higher diagonals are natural (Natural higher diagonal approximations).

[F6]

The higher diagonals preserve the chain complexes of subspaces (Natural higher diagonal approximations).

Proof

technique · compare two equivariant chain maps in the same explicit fourfold acyclic carrier

1.1given

Record the diagonal on the mod-two $C_{2}$ resolution. Let $W$ have one free-orbit generator $e_{i}$ in every degree $i \geq 0$ , with $d e_{i} = (1 + T) e_{i - 1}$ and $e_{- 1} = 0$ . Define

$ρ (e_{i}) = \sum_{r + s = i} e_{r} \otimes T^{r} e_{s} .$

With the diagonal $C_{2}$ action on $W \otimes W$ , this is a chain map. Indeed, expanding $d ρ (e_{i})$ makes every term with $r, s > 0$ occur twice after the index shifts $(r, s) \mapsto (r - 1, s)$ and $(r, s) \mapsto (r, s - 1)$ ; the two endpoint terms that remain are exactly $ρ ((1 + T) e_{i - 1})$ . All sums are finite.

1.2F4

Fix an explicit contraction of every common fourfold model carrier. For a standard simplex $Δ^{m}$ , let $P_{m}$ be the prism from its affine contraction to the first vertex. By [F4], $d P_{m} + P_{m} d = 1 - j_{m} p_{m}$ , where $p_{m}$ collapses to a point and $j_{m}$ includes that vertex. On the unnormalized point complex, whose degree- $n$ generator is $e_{n}$ , put $a (e_{n}) = e_{n + 1}$ for odd $n$ and $a (e_{n}) = 0$ for even $n$ . Directly, $d a + a d = 1 - η ϵ$ . Hence

$h_{m} : = P_{m} + j_{m} a p_{m}$

satisfies $d h_{m} + h_{m} d = 1 - j_{m} η ϵ p_{m}$ . On a tensor of four standard simplex complexes use $h_{1} \otimes 1 \otimes 1 \otimes 1 + e_{1} \otimes h_{2} \otimes 1 \otimes 1 + e_{1} \otimes e_{2} \otimes h_{3} \otimes 1 + e_{1} \otimes e_{2} \otimes e_{3} \otimes h_{4}$ , where each $e_{t}$ is its augmentation projection. The mixed terms cancel, giving a fixed $h^{(4)}$ with $d h^{(4)} + h^{(4)} d = 1 - η ϵ$ . Thus every positive-degree cycle and every augmentation-zero degree-zero cycle has the specified filling $h^{(4)} z$ . Every displayed operator is a finite sum, so this family of contractions is fixed without AC.

2.1F1F3F5step 1.1

Assemble the two displayed families into equivariant maps. Define $L (e_{i} \otimes z) = L_{i} (z)$ . This is the composite obtained by shuffling $z$ to $X \times Y$ , applying the equivariant higher diagonal there, and applying $A$ to its two outputs. Define $R$ by first applying $ρ$ , then applying the two higher-diagonal systems and finally regrouping with $τ$ . The factor $T^{r}$ in $R_{i}$ is precisely the twist in $ρ$ . Naturality and the chain-map identities in [F1]--[F3], together with step 1.1, show that both are $C_{2}$ -equivariant chain maps

$W \otimes (C \otimes E) ⟶ (C \otimes E)^{\otimes 2},$

where $C_{2}$ acts on the target by $Q$ .

3.1step 1.2step 2.1

Construct the coherent homotopy. Induct first on $i$ and then on $p + q$ . Suppose $H_{i - 1}$ and the values of $H_{i}$ on lower-dimensional model generators are known. On the identity model generator $z_{p, q}$ form

$ω = (L_{i} - R_{i}) (z_{p, q}) - H_{i} (d z_{p, q}) - (1 + Q) H_{i - 1} (z_{p, q}) .$

The chain-map equations for $L$ and $R$ from step 2.1 and the already established lower equations give $d ω = 0$ ; the two augmentation-preserving maps agree in total degree zero, so the remaining degree-zero case has augmentation zero. Set $H_{i} (z_{p, q}) = h^{(4)} ω$ and extend to arbitrary $u \otimes v$ by postcomposition. Taking its boundary gives exactly

$L_{i} - R_{i} = d H_{i} + H_{i} d + (1 + Q) H_{i - 1} .$

The postcomposition formula proves naturality. The fixed contractions and the lexicographic recursion use no choice principle.

4.1F2F3F6step 1.2step 3.1∎

The construction preserves the stated relative carriers. If $u$ lands in $B \subseteq X$ , every occurrence of $u_{#}$ in the two maps and in the model filling lands in $C_{*} (B)$ ; the other factor remains in $E$ . The same argument applies when $v$ lands in $D \subseteq Y$ . Linearity gives the assertions for their generated subcomplexes and for their sum. Empty factors give zero complexes; zero chains and $i = 0$ are included by $H_{- 1} = 0$ ; one-point and degenerate singular simplices remain in the unnormalized model. Hence every boundary case obeys the same equation.

Depends on

Dependency tree · 2 levels

Sources

Mosher and Tangora, Cohomology Operations and Applications