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The integral third Stiefel–Whitney classes of oriented tangent and normal bundles agree when the restricted ambient second class vanishes

Statement

Assume full AC. Let $Q$ and $Y$ be finite-dimensional Hausdorff second-countable smooth manifolds without boundary, and let $i : Q ↪ Y$ be a smooth embedding. Give $Y$ a smooth Riemannian metric and let $ν = (i^{*} T Y) / T Q$ be the real normal bundle, identified with the orthogonal complement of $T Q$ in $i^{*} T Y$ . Orient $T Q$ and $ν$ , and assume $w_{1} (T Q) = w_{1} (ν) = 0, w_{2} (i^{*} T Y) = 0.$ Define $W_{3} (E) : = {\tilde{β}}_{2} (w_{2} (E)) \in H^{3} (Q; Z)$ for an oriented real bundle $E$ , where ${\tilde{β}}_{2}$ is the integral Bockstein of $0 \to Z \overset{2}{\to} Z \to F_{2} \to 0$ . Then $W_{3} (T Q) = W_{3} (ν) .$

Facts & Assumptions

Given: Full AC, the oriented bundles $T Q, ν$ , their direct-sum identification with $i^{*} T Y$ , and $w_{2} (i^{*} T Y) = 0$ .

[F1]

Whitney sum formula for Stiefel–Whitney classes states $w (E \oplus F) = w (E) w (F)$ over the admissible bases of its statement.

[F2]

The first Stiefel–Whitney class classifies orientability states that an oriented numerable real bundle has $w_{1} = 0$ ; here the vanishing is also explicitly assumed.

[F3]

Stiefel–Whitney classes from the projective-bundle relation fixes the total classes, their degrees, and the rank conventions.

[F4]

Bockstein connecting operation defines ${\tilde{β}}_{2}$ from the displayed short exact coefficient sequence, and The Bockstein is independent of lift and representative proves that it is a well-defined function on cohomology classes.

[F5]

A finite-dimensional Hausdorff second-countable smooth manifold is paracompact Hausdorff, has CW homotopy type, and its finite-rank smooth vector bundles are numerable. Thus the base and bundle hypotheses of [F1] apply to $Q$ (Smooth manifolds have CW homotopy type).

Proof

Given: The bundle decomposition and vanishing classes above.

1.1F1F2F3F5givenalgebra

The base hypotheses of [F1] hold by [F5]. The degree-two part of the Whitney formula [F1] is $w_{2} (i^{*} T Y) = w_{2} (T Q) + w_{1} (T Q) w_{1} (ν) + w_{2} (ν) .$ Using the two orientation vanishings and $w_{2} (i^{*} T Y) = 0$ gives $w_{2} (T Q) + w_{2} (ν) = 0$ in $H^{2} (Q; F_{2})$ . Since this coefficient group has characteristic two, $w_{2} (T Q) = w_{2} (ν)$ .

2.1F3F4step 1.1algebra∎

Apply the same function ${\tilde{β}}_{2}$ of [F4] to the equal cohomology classes in step 1.1. By the definition of $W_{3}$ , the resulting equality is $W_{3} (T Q) = W_{3} (ν)$ in $H^{3} (Q; Z)$ .

Depends on

Dependency tree · 2 levels

Sources

John W. Milnor and James D. Stasheff, Characteristic Classes, §8