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Stiefel–Whitney classes from the projective-bundle relation

Definition

Assume AC, let $E \to B$ be a numerable real vector bundle of rank $n \geq 1$ over a paracompact Hausdorff CGWH base of CW homotopy type (an admissible base on this page), and let $x_{E} \in H^{1} (P (E); F_{2})$ be its tautological degree-one class. By Mod-two real projective bundle theorem there are unique classes $c_{i} \in H^{i} (B; F_{2})$ , $1 \leq i \leq n$ , with $x_{E}^{n} + c_{1} x_{E}^{n - 1} + \dots + c_{n} = 0 in H^{n} (P (E); F_{2}) .$ The Stiefel–Whitney classes of $E$ are these coefficients: $w_{i} (E) : = c_{i} \in H^{i} (B; F_{2}) (1 \leq i \leq n) .$ The definition is completed by the conventions $w_{0} (E) : = 1 \in H^{0} (B; F_{2})$ and $w_{i} (E) : = 0$ for $i > n$ , and the total Stiefel–Whitney class is the finite sum $w (E) : = \sum_{i \geq 0} w_{i} (E) = 1 + w_{1} (E) + \dots + w_{n} (E) \in H^{*} (B; F_{2}) .$ For the zero bundle of rank $0$ the conventions give $w (0_{B}) = 1$ .

Applying the definition to a line bundle $L \to B$ : here $n = 1$ , $P (L) ≅ B$ over $B$ and $γ_{L} ≅ L$ under that identification, so the relation is $x_{L} + w_{1} (L) = 0$ , that is, $w_{1} (L) = x_{L}$ , and $w (L) = 1 + x_{L}$ .

Facts & Assumptions

Given: AC, a numerable real rank- $n$ bundle $E \to B$ with $n \geq 1$ over a paracompact Hausdorff CGWH base of CW homotopy type, its projective bundle, and the class $x_{E}$ .

[F1]

Under AC, for a numerable positive-rank real bundle over a paracompact Hausdorff CGWH base of CW homotopy type, $H^{*} (P (E); F_{2})$ is free over $H^{*} (B; F_{2})$ on $1, x_{E}, \dots, x_{E}^{n - 1}$ , and there is a unique monic degree- $n$ relation $x_{E}^{n} + c_{1} x_{E}^{n - 1} + \dots + c_{n} = 0$ with $c_{i} \in H^{i} (B; F_{2})$ , which generates all polynomial relations (Mod-two real projective bundle theorem).

[F2]

For a rank-one bundle $L$ , the projection $P (L) \to B$ is a homeomorphism over $B$ and $γ_{L}$ corresponds to $L$ (Real projective bundle and tautological line).

[A1]

AC is the Axiom of Choice in the form fixed by The Axiom of Choice.

Verification

1.1

The classes $w_{i} (E)$ are well defined and have the asserted degrees and conventions. Existence and uniqueness of the coefficients $c_{i}$ is [F1], so each $w_{i} (E)$ is a single well-defined element of $H^{i} (B; F_{2})$ ; the relation is monic because its $x^{n}$ -coefficient is $1$ . The conventions $w_{0} = 1$ and $w_{i} = 0$ for $i > n$ extend the definition to all indices, and the total class is the finite sum of the nonzero terms, so it is a class in $H^{0} \oplus \dots \oplus H^{n}$ . For the zero bundle no positive coefficients exist and the total class is $1$ . The construction consumes AC only through [F1].

F1A1

2.1

The rank-one case. Let $L \to B$ be a numerable real line bundle. By [F2], $P (L) ≅ B$ over $B$ and the tautological line $γ_{L}$ is $L$ , so $x_{L} \in H^{1} (P (L); F_{2}) = H^{1} (B; F_{2})$ and the defining relation of [F1] reads $x_{L} + w_{1} (L) = 0$ in $H^{1} (B; F_{2})$ . Since $- 1 = 1$ in $F_{2}$ , this gives $w_{1} (L) = x_{L}$ ; the conventions give $w_{i} (L) = 0$ for $i \geq 2$ and $w (L) = 1 + x_{L}$ .

F1F2step 1.1algebra

3.1

Boundary cases. In rank one the fiber ${R P}^{0}$ is a point, so the base of the relation is the whole base $B$ and the displayed computation is literal. Over the empty base every group is zero, the relation is the zero relation, and the conventions give the zero classes with $w_{0} = 1 = 0$ in the zero ring. The rank-zero convention $w (0_{B}) = 1$ is the unit, matching the degree-zero convention used in every rank.

F1F2step 1.1step 2.1∎

Depends on

Used by

An odd-rank Euler class need not vanish in the presence of two-torsion Counterexample
Real flag bundle and Stiefel–Whitney roots Definition
Stiefel–Whitney class of the universal real line Example
Integral powers of the complexified universal real line Lemma
The first Stiefel–Whitney class classifies orientability Proposition
Mod-two cohomology of BO(n) Theorem
Mod-two reduction of Chern classes Theorem
Naturality of Stiefel–Whitney classes Theorem
Naturality, stability, and mod-two reduction of Pontryagin classes Theorem
The mod-two Euler class is the top Stiefel–Whitney class Theorem
Thom identity for Stiefel–Whitney classes Theorem
Uniqueness of Stiefel–Whitney classes from normalization, naturality, and sum Theorem
Whitney sum formula for Stiefel–Whitney classes Theorem

Dependency tree · two levels

21 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

Allen Hatcher, Vector Bundles & K-Theory (standard reference, not scraped)
Haynes Miller, MIT 18.906 Algebraic Topology II lecture notes (standard reference, not scraped)
Milnor and Stasheff, Characteristic Classes (standard reference, not scraped)