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Boundaries have zero Stiefel-Whitney numbers

Statement

Assume AC (The Axiom of Choice), used only through the Stiefel-Whitney class construction, its admissibility input, and the inward field used in the boundary tangent splitting (which requires ${A C}_{ω}$ ). Let $M$ be a closed smooth $n$ -manifold that is the boundary of a compact smooth $(n + 1)$ -manifold $W$ , with canonical mod-two fundamental class $[M] \in H_{n} (M; F_{2})$ (Stiefel-Whitney numbers of a closed manifold) and inclusion $i : M ↪ W$ . Then $w^{I} [M] = ⟨ w^{I} (T M), [M] ⟩ = 0$ for every monomial $w^{I} = w_{1}^{r_{1}} \dots w_{n}^{r_{n}}$ of total degree $n$ . Hence every Stiefel-Whitney number of a closed boundary vanishes, and a closed manifold with at least one nonzero Stiefel-Whitney number is not null-cobordant (Null-cobordant closed manifolds).

Facts & Assumptions

Given: A closed smooth $n$ -manifold $M = \partial W$ that is the boundary of a compact smooth $(n + 1)$ -manifold $W$ , its inclusion $i : M \to W$ , and the monomials $w^{I}$ of total degree $n$ in the tangent classes of $M$ . AC is assumed.

[F1]

The restriction of the tangent bundle of $W$ to the boundary splits off a trivial line: $T W ∣_{M} ≅ T M \oplus ε^{1}$ , under ${A C}_{ω}$ , which follows from AC (The boundary stable tangent bundle splits off a trivial line, The Axiom of Choice).

[F2]

Stiefel-Whitney classes are natural, $w_{i} (f^{*} E) = f^{*} w_{i} (E)$ , and satisfy the Whitney sum formula with stability for trivial summands, $w (E \oplus ε^{r}) = w (E)$ ; the construction applies to $M$ because a closed smooth manifold is a paracompact Hausdorff CGWH base of CW homotopy type with numerable tangent bundle (Naturality of Stiefel–Whitney classes, Whitney sum formula for Stiefel–Whitney classes, Stiefel–Whitney classes from the projective-bundle relation, Smooth manifolds have CW homotopy type).

[F3]

The Kronecker pairing is natural: $⟨ i^{*} α, z ⟩ = ⟨ α, i_{*} z ⟩$ , and is additive (Kronecker evaluation pairing, The kronecker pairing is independent of cocycle and cycle representatives).

[F4]

The fundamental class of the boundary pushes forward to zero: $i_{*} [M] = 0$ in $H_{n} (W; F_{2})$ , where $[M]$ is the fundamental class of the canonical mod-two orientation (The fundamental class of a boundary pushes forward to zero, Every manifold is F2-orientable and orientability is componentwise, Fundamental class of a compact oriented manifold).

[F5]

A Stiefel-Whitney number of a closed smooth $n$ -manifold is the evaluation $w^{I} [M] = ⟨ w^{I} (T M), [M] ⟩$ of a degree- $n$ monomial, it is a diffeomorphism invariant, and $M$ is null-cobordant exactly when $M$ is diffeomorphic to the whole boundary of a compact smooth $(n + 1)$ -manifold (Stiefel-Whitney numbers of a closed manifold, Null-cobordant closed manifolds).

Proof

1.1F1F2

(The tangent classes of $M$ are restrictions from $W$ .) By [F1], $T W ∣_{M} ≅ T M \oplus ε^{1}$ . Applying naturality and the Whitney sum formula with the trivial summand from [F2] gives $w (T M) = w (T M \oplus ε^{1}) = w (T W ∣_{M}) = w (i^{*} T W) = i^{*} w (T W),$ and hence $w^{I} (T M) = i^{*} w^{I} (T W)$ for every monomial.

2.1F3F4step 1.1

(The evaluations vanish.) For a degree- $n$ monomial $w^{I}$ , naturality of the Kronecker pairing [F3] and the vanishing of the boundary pushforward [F4] give $w^{I} [M] = ⟨ w^{I} (T M), [M] ⟩ = ⟨ i^{*} w^{I} (T W), [M] ⟩ = ⟨ w^{I} (T W), i_{*} [M] ⟩ = ⟨ w^{I} (T W), 0 ⟩ = 0.$

3.1F5step 2.1∎

(All numbers vanish; null-cobordism consequence.) Since the monomial $w^{I}$ of total degree $n$ was arbitrary, every Stiefel-Whitney number of the closed boundary $M = \partial W$ vanishes. If a closed smooth $n$ -manifold $N$ is null-cobordant, then by [F5] it is diffeomorphic to the whole boundary of some compact smooth $(n + 1)$ -manifold, and diffeomorphism invariance of the numbers transfers the vanishing to $N$ . Contrapositively, a closed manifold with at least one nonzero Stiefel-Whitney number is not null-cobordant.

Depends on

Used by

The real projective plane is not unoriented null-cobordant Counterexample

Dependency tree · two levels

80 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

John Milnor and James Stasheff, Characteristic Classes (original pagination) (standard reference, not scraped)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, 2002) (standard reference, not scraped)