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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Sq^1 is the mod-two Bockstein

Statement

For every space $X$ , every $n \geq 0$ , and every $x \in H^{n} (X; F_{2})$ ,

$S q^{1} (x) = β (x),$

where $β$ is the Bockstein of

$0 ⟶ F_{2} \overset{2}{\to} Z / 4 ⟶ F_{2} ⟶ 0.$

This equality, including its residue-lift calculation, requires no AC.

Facts & Assumptions

Given: A space $X$ , a nonnegative degree $n$ , and $x \in H^{n} (X; F_{2})$ .

[F1]

The displayed cyclic coefficient sequence defines the mod-two Bockstein, and least residue representatives supply its lifts without AC (Bockstein connecting operation).

[F2]

The choice-free parity recurrence gives $β S q^{j} = S q^{j + 1}$ when $j$ is even (Bockstein parity recurrence for Steenrod squares).

[F3]

The zero square is the identity in every degree (Steenrod normalization, instability, suspension, and top square).

Proof

Proof technique: specialize the proved Bockstein recurrence at the zero square.

1.1

Apply the parity recurrence at $j = 0$ .

givenF2F3

$β (x) = β S q^{0} (x) = S q^{1} (x) .$

This is the claimed equality of operations.

2.1

Both operations vanish on the empty space and zero class, and the canonical lifts use no choice. [F1, F2, F3, step 1.1] For the empty space or zero class, both sides vanish. On a point and, more generally, in degree zero, [F3] makes $S q^{1}$ zero by instability, so step 1.1 makes the Bockstein zero as well. The index $j = 0$ is included explicitly in [F2], and no negative degree or converse assertion occurs. Ordinary unnormalized singular cochains, including degenerate simplices, are inherited from [F1] and [F2]. The only lift used in [F2] is the valuewise zero/one residue lift described in [F1], so the equality is choice-free and assumes no AC. ∎

Depends on

Used by

Steenrod squares do not all lift integrally Counterexample
Bockstein detects integral two-torsion in real projective space Example
The relation Sq¹Sq¹=0 Example
Wu classes of a closed surface Example

Dependency tree · two levels

14 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 in Homotopy Theory (standard reference, not scraped)