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

How statement and proof provenance work

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.

Finite-cellular cyclic squares agree with singular cup-i squares

Statement

Assume AC. Let $K$ be a finite oriented regular cell complex. There is a cell-carried chain map

$ι : C_{*}^{c e l l} (K; F_{2}) ⟶ C_{*}^{s i n g} (K; F_{2})$

whose dual induces an isomorphism

$ι^{*} : H_{s i n g}^{*} (K; F_{2}) \overset{≅}{\to} H_{c e l l}^{*} (K; F_{2}) .$

For $y \in H_{s i n g}^{q} (K; F_{2})$ and every integer $i$ ,

$ι^{*} S q^{i} (y) = S q_{c y c}^{i} (ι^{*} y) .$

Here the left square is the singular cup- $i$ square and the right square is the finite-cellular cyclic-power square. The equality is unchanged if one replaces $ι$ , the carried cellular diagonal, or the carried singular higher diagonal by another comparison of the stated kind. This lemma makes the comparison only on finite regular complexes; it does not extend the cyclic construction to arbitrary spaces.

Facts & Assumptions

Given: AC, a finite oriented regular cell complex $K$ , and mod-two cellular and ordinary unnormalized singular chains.

[F1]

A carried equivariant cellular diagonal computes the cyclic coefficient $D_{j}$ (Equivariant p-fold external power and diagonal decomposition).

[F2]

Carried equivariant chain maps extend and are homotopy-unique relative to the subcomplex where they were fixed (Equivariant p-fold external power and diagonal decomposition).

[F3]

The finite-cellular operation is $S q_{c y c}^{i} (x) = D_{q - i} (x)$ in degree $q$ , with zero outside $0 \leq i \leq q$ (Finite-cellular cyclic squares, Cartan formula, and cyclic-basis action).

[F4]

The singular higher diagonals satisfy $d D_{r} + D_{r} d = (1 + T) D_{r - 1}$ , preserve subspaces, and are coherently unique (Natural higher diagonal approximations).

[F5]

The singular cup- $i$ definition represents $S q^{i} [y]$ by $a ⌣_{q - i} a$ for a degree- $q$ cocycle $a$ (Steenrod squares from cup-i).

[F6]

Cellular homology agrees with singular homology on a CW complex (Cellular homology computes singular homology).

[F7]

Alexander--Whitney and shuffle are augmentation-preserving chain-homotopy inverses on ordinary unnormalized chains (Alexander--Whitney and shuffle are natural chain-homotopy inverses).

[F8]

A homotopy equivalence induces an isomorphism on singular homology (Homotopy equivalences induce isomorphisms on singular homology).

[F9]

AC supplies choice functions for arbitrary families of nonempty sets (The Axiom of Choice).

Proof

Proof technique: compare the cellular and singular equivariant diagonals inside one acyclic carrier, then evaluate the resulting chain homotopy on a cocycle.

1.1

Construct a chain comparison carried by closed cells. [given, F6] The first barycentric subdivision of a finite regular cell complex is a finite simplicial complex. For each $n$ -cell $e$ , let $s (e)$ be the mod-two sum of the oriented $n$ -simplices subdividing its closed ball. The codimension-one faces internal to $\overline{e}$ occur twice and cancel, while the remaining faces occur with precisely the cellular incidence coefficients. Hence $s (d e) = d s (e)$ . Including these simplicial chains as singular chains defines $ι$ . Its value on $e$ is supported in $\overline{e}$ , so it is cell-carried. The relative fundamental simplex in each pair $(K^{n}, K^{n - 1})$ maps to the same relative fundamental class; thus the induced map is the standard cellular-to-singular comparison of [F6] and is an isomorphism on homology.

1.2

Prove that the dual comparison is an isomorphism and locate its choice cost. [F9, step 1.1] Let $Q$ be the mapping cone of $ι$ . Step 1.1 says that $Q$ is acyclic. For every $n$ , [F9] chooses a complement $L_{n}$ to $B_{n} (Q) = Z_{n} (Q)$ in $Q_{n}$ . The differential restricts to an isomorphism $d : L_{n} \to B_{n - 1} (Q)$ . Define $h$ to be its inverse on $B_{n - 1} (Q)$ and zero on $L_{n - 1}$ . On the decomposition $Q_{n} = B_{n} (Q) \oplus L_{n}$ one checks directly that $d h + h d = 1_{Q}$ . Dualizing this identity contracts $Hom (Q, F_{2})$ , which is the shifted mapping cone of $ι^{*}$ . Therefore $ι^{*}$ is an isomorphism on cohomology. This use of AC is needed because the singular chain spaces and the family of complements need not be finite.

2.1

Package both systems as equivariant carried chain maps. [F1, F4, F7, F8, step 1.1] Let $W$ be the standard free $F_{2} [C_{2}]$ -resolution with $d e_{r} = (1 + T) e_{r - 1}$ . By [F1], choose a cell-carried equivariant diagonal

$Φ_{C} : W \otimes C_{*}^{c e l l} (K) ⟶ C_{*}^{c e l l} (K) \otimes C_{*}^{c e l l} (K) .$

By [F4], the formula $Φ_{S} (e_{r} \otimes z) = D_{r} (z)$ defines an equivariant chain map

$Φ_{S} : W \otimes C_{*}^{s i n g} (K) ⟶ C_{*}^{s i n g} (K) \otimes C_{*}^{s i n g} (K),$

because its chain-map equation is exactly $d D_{r} + D_{r} d = (1 + T) D_{r - 1}$ . For a cell $e$ , both $(ι \otimes ι) Φ_{C}$ and $Φ_{S} (1 \otimes ι)$ send $W \otimes e$ into $C_{*}^{s i n g} (\overline{e})^{\otimes 2}$ . The closed cell is a disk; [F8] makes its augmented singular complex acyclic, and [F7] identifies the tensor target up to augmentation-preserving chain homotopy with the singular chains of its square. Thus these targets form one equivariant augmented-acyclic carrier.

3.1

Compare the two diagonals in that carrier. [F2, F9, step 2.1] Both maps in step 2.1 preserve the degree-zero augmentation and are carried by the same closed-cell diagonal carrier. The relative equivariant carrier comparison in [F2] supplies an equivariant chain homotopy $H$ with

$Φ_{S} (1 \otimes ι) - (ι \otimes ι) Φ_{C} = d H + H d .$

Over $F_{2}$ subtraction is addition. The AC expenditure in this step is exactly [F9]'s selection of one orbit representative and one filling in each nonempty carrier-extension problem, as already isolated in [F2].

4.1

Evaluate the comparison and identify every square. [F1, F2, F3, F4, F5, step 1.2, step 3.1] Let $a$ be a singular degree- $q$ cocycle representing $y$ , and put $c = a ι$ . For $0 \leq i \leq q$ , set $r = q - i$ . By [F5], the pullback of the singular square is represented on a cellular chain $z$ by

$z ⟼ (a \otimes a) Φ_{S} (e_{r} \otimes ι z) .$

By [F1] and [F3], the cyclic square is represented by

$z ⟼ (c \otimes c) Φ_{C} (e_{r} \otimes z) = (a \otimes a) (ι \otimes ι) Φ_{C} (e_{r} \otimes z) .$

Evaluate the homotopy identity of step 3.1 by the invariant cocycle $a \otimes a$ . The resulting two equivariant cochains on $W \otimes C_{*}^{c e l l} (K)$ differ by the coboundary of $(a \otimes a) H$ . Since $d e_{r} = (1 + T) e_{r - 1}$ and the evaluated cochain is $C_{2}$ -invariant, its $W$ -differential is zero. Consequently the $e_{r}$ coordinates displayed above differ by an ordinary cellular coboundary. Their cohomology classes are equal, which is the asserted formula. Coherent uniqueness in [F4] and the same carrier homotopy in [F2] prove independence of every stated comparison choice.

5.1

Check ranges, degeneracies, and choices. [F3, F4, F5, F9, step 1.1, step 1.2, step 4.1] For the empty complex all chain groups vanish. The zero class is represented by the zero cocycle, and on a point the only nonzero assertion is $S q^{0} = S q_{c y c}^{0} = i d$ in degree zero. The indices $i = 0$ and $i = q$ correspond respectively to $e_{q}$ and $e_{0}$ ; both occur in the evaluation step, while $i < 0$ and $i > q$ are zero on both sides by [F3] and [F5]. Ordinary unnormalized singular chains are used throughout, and [F4] includes every degenerate singular simplex, so no normalization quotient is hidden. Barycentric subdivision and all sums within a fixed finite $K$ are finite. AC is used only for the set-indexed carrier fillings in step 3.1 and the vector-space complements in step 1.2. No arbitrary-space extension, converse, or Adem relation is used. ∎

Depends on

Used by

Adem relations for Steenrod squares Theorem

Dependency tree · two levels

36 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

N. E. Steenrod and D. B. A. Epstein, Cohomology Operations (standard reference, not scraped)
Mosher and Tangora, Cohomology Operations and Applications (standard reference, not scraped)