LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

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.

First Steenrod square relation for w2

Statement

Assume full AC. Let $E \to B$ be a numerable finite-rank real vector bundle, meaning it has a linear trivializing cover with a locally finite subordinate partition of unity. Let $B$ be paracompact and Hausdorff, compactly generated (a subset is closed exactly when every continuous map from a compact Hausdorff space has closed inverse image of it), and weak Hausdorff (every continuous map from a compact Hausdorff space has closed image); also require $B$ to be homotopy equivalent to a CW complex. With Stiefel–Whitney classes in $H^{*} (B; F_{2})$ and Steenrod squares in singular cohomology, one has $S q^{1} w_{2} (E) = w_{1} (E) ⌣ w_{2} (E) + w_{3} (E) .$ The conventions $w_{j} (E) = 0$ for $j$ above the rank include ranks zero and one.

Statement references

Stiefel–Whitney classes from the projective-bundle relation

Real splitting principle with mod-two injective pullback

Compactly generated conventions for based homotopy

Real and complex topological vector bundles

Hatcher, Vector Bundles & K-Theory, Proposition 3.3

Miller, Algebraic Topology II, Lecture 34

Facts & Assumptions

Given: A numerable finite-rank real vector bundle on the stated admissible base; assume the full AC required by the cited splitting/class construction inputs.

[F1]

Real splitting principle with mod-two injective pullback: the real flag pullback $q^{*}$ on mod-two cohomology is injective and $q^{*} E = ⨁_{j} L_{j}$ .

[F2]

Real flag bundle and Stiefel–Whitney roots: set $t_{j} = w_{1} (L_{j}) \in H^{1} (Fl (E); F_{2})$ for the split line summands.

[F3]

Whitney sum formula for Stiefel–Whitney classes: $w (⨁_{j} L_{j}) = \prod_{j} w (L_{j}) = \prod_{j} (1 + t_{j})$ , so $w_{1} = \sum_{j} t_{j}$ , $w_{2} = \sum_{i < j} t_{i} t_{j}$ , and $w_{3} = \sum_{i < j < k} t_{i} t_{j} t_{k}$ after pullback.

[F4]

Naturality of Stiefel–Whitney classes: the classes $w_{i}$ commute with bundle pullback.

[F5]

Steenrod squares are well-defined and natural: $S q^{1}$ is a natural cohomology operation.

[F6]

Steenrod normalization, instability, suspension, and top square: for a degree-one class $t$ , $S q^{1} (t) = t^{2}$ .

[F7]

Cartan formula for Steenrod squares: $S q^{1} (x y) = S q^{1} (x) y + x S q^{1} (y)$ .

[F8]

Cup-i coboundary identity: the cup-one coboundary identity implies that, for mod-two cocycles, the two orders of the cup product represent the same cohomology class; hence the mod-two cohomology product is commutative.

[F9]

Cup product is natural, unital and associative: every continuous pullback is a ring map, so $q^{*} (x y) = q^{*} x q^{*} y$ .

[A1]

Full AC is assumed as defined in The Axiom of Choice, exactly as required by [F1] and the Stiefel–Whitney construction.

Proof

1.1F1F2F3F4F5F6F7A1

On the flag bundle, write $q^{*} E = L_{1} \oplus \dots \oplus L_{n}$ and $t_{i} = w_{1} (L_{i})$ by [F1]–[F2]. The Whitney formula [F3] gives $q^{*} w_{2} (E) = \sum_{i < j} t_{i} t_{j}$ using naturality [F4]. Apply [F7] to each summand and use [F6]: $S q^{1} q^{*} w_{2} (E) = \sum_{i < j} (t_{i}^{2} t_{j} + t_{i} t_{j}^{2}) .$ By [F5], this equals $q^{*} S q^{1} w_{2} (E)$ .

2.1F1F3F4F8F9A1step 1.1algebra

By [F9], $q^{*} (w_{1} (E) w_{2} (E)) = q^{*} w_{1} (E) q^{*} w_{2} (E) = (\sum_{k} t_{k}) (\sum_{i < j} t_{i} t_{j})$ , using [F3]. Reorder factors using mod-two cup-product commutativity [F8]. Terms for which $k = i$ or $k = j$ give exactly $\sum_{i < j} (t_{i}^{2} t_{j} + t_{i} t_{j}^{2})$ , the expression in step 1.1. For each triple of distinct indices $i < j < k$ , the monomial $t_{i} t_{j} t_{k}$ occurs three times, once for each choice of the singled-out factor $t_{ℓ}$ in $w_{1}$ ; three equals one in $F_{2}$ . Therefore $q^{*} (w_{1} w_{2}) = S q^{1} q^{*} w_{2} + \sum_{i < j < k} t_{i} t_{j} t_{k} = S q^{1} q^{*} w_{2} + q^{*} w_{3} .$ The empty pair sum for ranks zero and one and the empty triple sum for ranks below three obey the same expansion.

3.1F1A1step 2.1algebra∎

Rearranging the mod-two identity in step 2.1 gives $q^{*} (S q^{1} w_{2} + w_{1} w_{2} + w_{3}) = 0$ . Since $q^{*}$ is injective by [F1] and subtraction equals addition over $F_{2}$ , the class inside the pullback is zero. This is the asserted identity for all ranks in the stated scope.

First Steenrod square relation for w2

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources