DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Mod-p reduced power operations

Definition

Assume AC, let $p$ be an odd prime, and put $m = (p - 1) / 2$ . For $q \geq 0$ , $x \in H^{q} (X; F_{p})$ , and $i \geq 0$ , define

P^{i} (x) : = (- 1)^{i + m (q^{2} + q) / 2} (m!)^{- q} D_{(q - 2 i) (p - 1)} (x) \in H^{q + 2 i (p - 1)} (X; F_{p}) .

Here $D_{j}$ is the cyclic coefficient operation of Cyclic p-fold power construction, and the negative exponent denotes the inverse of the nonzero element $(m!)^{q} \in F_{p}^{\times}$ . Define $P^{i} = 0$ for $i < 0$ , and define

β P^{i} : = β \circ P^{i} : H^{q} (X; F_{p}) ⟶ H^{q + 2 i (p - 1) + 1} (X; F_{p})

using the mod- $p$ Bockstein associated to $0 \to Z / p \overset{p}{\to} Z / p^{2} \to Z / p \to 0$ . For $i < 0$ , set $β P^{i} = 0$ as well.

The inverse factorial is intentional. The displayed formula in Steenrod--Epstein VII Definition 6.1 prints $(m!)^{q}$ , but its proof of Lemma 6.4 uses $(m!)^{- q}$ . The inverse is forced by the top coefficient in the preceding lemma; for example, at $p = 5, q = 1$ , the printed positive power would make $P^{0} = - i d$ .

Facts & Assumptions

Given: AC, an odd prime $p$ , $m = (p - 1) / 2$ , a space $X$ , a degree $q \geq 0$ class, and an integer $i$ .

[F1]

The cyclic coefficients are natural and additive and vanish for $j < 0$ or $j > (p - 1) q$ (Cyclic p-fold power construction).

[F2]

The mod- $p$ Bockstein is the connecting operation for the cyclic coefficient sequence (Bockstein connecting operation).

[F3]

The Bockstein is independent of its lift and cocycle representative (The Bockstein is independent of lift and representative).

[F4]

The top cyclic coefficient is $D_{(p - 1) q} (x) = (- 1)^{m q (q + 1) / 2} (m!)^{q} x$ (Cyclic p-fold power construction).

[A1]

The Axiom of Choice is assumed exactly because the singular cyclic coefficient supplier [F1] assumes it.

Verification

Proof technique: check the grading and normalization directly from the cyclic coefficient formula.

1.1

The formula is defined and has the stated degree. [given, F1, A1] None of $1, \dots, m$ is zero in $F_{p}$ , so $m!$ and every $(m!)^{q}$ are units. If $j = (q - 2 i) (p - 1)$ , then

p q - j = p q - (q - 2 i) (p - 1) = q + 2 i (p - 1) .

Thus the scalar multiple of $D_{j} (x)$ lies in the displayed target. Since $q (q + 1)$ is even, the sign exponent is an integer. Naturality and additivity are inherited from [F1].

2.1

The normalization gives $P^{0} = i d$ . [F4, step 1.1] At $i = 0$ , [F4] gives

P^{0} (x) = (- 1)^{m q (q + 1) / 2} (m!)^{- q} (- 1)^{m q (q + 1) / 2} (m!)^{q} x = x .

The two equal sign exponents add to an even integer, and the factorial factors cancel.

2.2

The index conventions include negative operations and instability. [F1, step 1.1] For $i < 0$ , the operation is zero by definition; equivalently its cyclic index exceeds $(p - 1) q$ . If $i \geq 0$ and $2 i > q$ , then $(q - 2 i) (p - 1) < 0$ , so [F1] makes $P^{i} (x) = 0$ . At $2 i = q$ , the cyclic index is zero and the formula legitimately uses $D_{0} (x) = x^{p}$ ; it is not included in the vanishing range.

3.1

The Bockstein composite and all boundary cases are well-defined. [F2, F3, A1, step 1.1, step 2.1, step 2.2] The specified cyclic short exact sequence and [F2] define the positive mod- $p$ Bockstein, while [F3] makes the resulting cohomology operation independent of cochain choices. Hence its composite with $P^{i}$ has degree one more than $P^{i}$ .

For the empty space and the zero class, both operations are zero. At $q = 0$ , step 2.1 gives $P^{0} = i d$ , while every $i > 0$ is in the strict instability range; this includes the point and its elements zero and one. The endpoints $i = 0$ and $2 i = q$ , negative $i$ , and $2 i > q$ are all explicit. Degenerate singular simplices require no new convention because the operations are formed by composing the already well-defined suppliers. No biconditional is asserted. This definition makes no new selection: AC is propagated exactly from [F1], and the cyclic Bockstein uses canonical least residue lifts. ∎

Depends on

Used by

Reduced powers satisfy naturality, instability, Cartan, and Adem relations Theorem

Dependency tree · two levels

16 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)