DefinitionDefinition: 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.

Wu classes of a closed manifold

Definition

Assume AC. Let $M$ be a closed topological $n$ -manifold, possibly empty or disconnected. Give it its canonical mod-two orientation and write $[M]_{2}$ for the resulting fundamental class. For $0 \leq k \leq n$ , the $k$ th Wu class is the unique class $v_{k} (M) \in H^{k} (M; F_{2})$ such that

⟨ v_{k} (M) ⌣ x, [M]_{2} ⟩ = ⟨ S q^{k} (x), [M]_{2} ⟩ for every x \in H^{n - k} (M; F_{2}) .

Set $v_{k} (M) = 0$ outside $0 \leq k \leq n$ . The total Wu class is the finite sum

v (M) : = \sum_{k = 0}^{n} v_{k} (M) \in H^{*} (M; F_{2}) .

Facts & Assumptions

Given: The closed $n$ -manifold $M$ and an integer $k$ .

[F1]

Every manifold has a canonical $F_{2}$ -orientation, and a compact manifold has finitely many components (Every manifold is F2-orientable and orientability is componentwise).

[F2]

Assuming AC, the mod-two cup pairing of a closed oriented manifold is perfect in both variables (Poincaré duality gives a nonsingular cup pairing).

[F3]

Each $S q^{k}$ is an additive cohomology operation (Steenrod squares are well-defined and natural).

[F4]

On $H^{d} (-; F_{2})$ , $S q^{0}$ is the identity and $S q^{k}$ is zero for $k > d$ (Steenrod normalization, instability, suspension, and top square).

[A1]

The Axiom of Choice is assumed exactly because [F2] assumes it; no new family of choices is made here.

Verification

Proof technique: represent the Steenrod-square functional by the perfect Poincaré cup pairing.

1.1

Fix $0 \leq k \leq n$ . [F1, F3] The canonical orientation supplies $[M]_{2}$ . Since $S q^{k}$ is additive, the map

L_{k} ⁣ : H^{n - k} (M; F_{2}) ⟶ F_{2}, L_{k} (x) = ⟨ S q^{k} (x), [M]_{2} ⟩

is an $F_{2}$ -linear functional. This remains true componentwise: the fundamental class is the finite sum of the component classes and evaluation is additive.

1.2

The first adjoint of the cup pairing is an isomorphism. [F2, A1] In the present degrees it is

H^{k} (M; F_{2}) \overset{≅}{\to} {Hom}_{F_{2}} (H^{n - k} (M; F_{2}), F_{2}), a ⟼ (x ⟼ ⟨ a ⌣ x, [M]_{2} ⟩) .

Consequently $L_{k}$ has exactly one preimage. Defining that preimage to be $v_{k} (M)$ proves both existence and uniqueness in the displayed definition. AC is used only through the already proved perfectness assertion [F2].

1.3

The normalization and high-degree components are determined. [F2, F4, step 1.2] For $k = 0$ , [F4] gives $L_{0} (x) = ⟨ x, [M]_{2} ⟩$ . The unit $1 \in H^{0} (M; F_{2})$ represents this functional, so uniqueness gives $v_{0} (M) = 1$ . If $2 k > n$ , every $x \in H^{n - k}$ has degree $n - k < k$ ; instability in [F4] makes $S q^{k} (x) = 0$ . Thus $L_{k} = 0$ , and injectivity of the adjoint gives $v_{k} (M) = 0$ . This includes $k = n > 0$ .

2.1

For the empty manifold the Wu classes and defining functionals vanish, with degree-zero unit equal to zero. [F1, F2, F3, F4, A1, step 1.1, step 1.2, step 1.3] For the empty manifold all displayed groups and functionals are zero, and the degree-zero unit is the zero element of its zero cohomology ring. For a point, $n = 0$ and $v = v_{0} = 1$ . The zero functional is represented by the zero class. The endpoints $k = 0, n$ were treated in step 1.3; indices $k < 0$ and $k > n$ are zero by the stated convention, so the total sum is finite. Disconnected manifolds are included by the finite component sum in step 1.1. Degenerate singular simplices require no new convention because [F2] and [F3] are statements about ordinary singular cohomology. No biconditional is asserted. Apart from the AC already exposed by [F2], the definition makes no choice. ∎

Depends on

Used by

Wu classes of a closed surface Example

Dependency tree · two levels

26 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

Husemöller, Joachim, Jurčo, and Schottenloher, Basic Bundle Theory and K-Cohomology Invariants (standard reference, not scraped)