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Third Stiefel–Whitney Classes of Tangent and Normal Bundles

Statement

Let $X$ be a paracompact Hausdorff compactly generated weak Hausdorff space of CW homotopy type, and let $E, F \to X$ be numerable finite-rank real vector bundles. Assume full AC as required by the imported characteristic-class and Bockstein suppliers. If

w_{1} (E) = w_{1} (F) = 0, w_{2} (E \oplus F) = 0

in mod-two singular cohomology, then

W_{3} (E) = W_{3} (F) and 2 W_{3} (E) = 0

in $H^{3} (X; Z)$ . In particular, the equality applies to an oriented tangent bundle and its oriented normal bundle when their direct sum has vanishing $w_{2}$ .

Statement references

Integral Third Stiefel–Whitney Class

Whitney sum formula for Stiefel–Whitney classes

Hatcher, Vector Bundles & K-Theory, Chapter 3 §3.1

Miller, Algebraic Topology II, Lectures 34–36

Facts & Assumptions

Given: Full AC; numerable finite-rank real bundles $E, F \to X$ over the stated base, with $w_{1} (E) = w_{1} (F) = 0$ and $w_{2} (E \oplus F) = 0$ .

[F1]

Stiefel–Whitney classes from the projective-bundle relation: $w_{j} (G) \in H^{j} (X; Z / 2)$ are the Stiefel–Whitney classes of a numerable finite-rank real bundle $G$ .

[F2]

Whitney sum formula for Stiefel–Whitney classes: $w (E \oplus F) = w (E) ⌣ w (F)$ .

[F3]

Integral Third Stiefel–Whitney Class and Bockstein connecting operation: $W_{3} (G) = {\tilde{β}}_{2} (w_{2} (G))$ , with ${\tilde{β}}_{2}$ the integral connecting operation for $0 \to Z \overset{\times 2}{\to} Z \to Z / 2 \to 0$ .

[F4]

Bockstein connecting operation and Singular cohomology with coefficients: the connecting homomorphism is computed by lifting a mod-two singular cocycle to an integral cochain and dividing its integral coboundary by $2$ .

[F5]

The Bockstein is independent of lift and representative: this operation gives a well-defined cohomology class, independent of the cochain lift and cocycle representative.

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function; this is the full choice hypothesis of the imported Bockstein and characteristic-class suppliers.

Proof

1.1givenF1F2A1algebra

The degree-two part of [F2] gives $w_{2} (E \oplus F) = w_{2} (E) + w_{1} (E) ⌣ w_{1} (F) + w_{2} (F)$ . The two degree-one classes vanish by the Given clause, and the left side vanishes by the Given clause, so $w_{2} (E) + w_{2} (F) = 0$ . Since these classes have coefficients in $Z / 2$ , this says $w_{2} (E) = w_{2} (F)$ .

2.1F3F5A1step 1.1algebra

Apply the well-defined operation ${\tilde{β}}_{2}$ to the equality in step 1.1 and use [F3]. This gives $W_{3} (E) = W_{3} (F)$ .

3.1givenF3F4F5A1algebra∎

Represent $w_{2} (E)$ by a mod-two singular 2-cocycle $c$ . Lift each value of $c$ to its unique representative $0$ or $1$ , obtaining an integral 2-cochain $\tilde{c}$ . Because $c$ is a cocycle, every value of $δ \tilde{c}$ is even, so $d = (δ \tilde{c}) / 2$ is an integral 3-cochain. By [F4] and the lift-independence in [F5], $W_{3} (E) = [d]$ . Therefore $2 W_{3} (E) = [2 d] = [δ \tilde{c}] = 0$ , since $δ \tilde{c}$ is a coboundary.

Remarks

The equality uses the integral Bockstein, so torsion is retained. The proof does not assume a classification theorem for spin structures; its input is the explicit vanishing of $w_{2} (E \oplus F)$ .

Third Stiefel–Whitney Classes of Tangent and Normal Bundles

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources