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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

1.1F1A1

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].

2.1F1F2step 1.1algebra

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 $w (L) = 1 + x_{L}$ .

3.1F1F2step 1.1step 2.1∎

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.

Stiefel–Whitney classes from the projective-bundle relation

Definition

Facts & Assumptions

Verification

Depends on

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