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Real flag bundle and Stiefel–Whitney roots

Definition

Assume AC, and 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. The real flag bundle of $E$ is obtained by the following finite iteration.

At the first stage put $X_{1} : = P (E)$ , let $q_{1} : X_{1} \to B$ be the projection, let $E_{1} : = q_{1}^{*} E$ , and let $L_{1} \subseteq E_{1}$ be the tautological line. By Numerable vector bundles admit bundle metrics choose a bundle metric on the numerable bundle $E_{1}$ , and let $L_{1}^{⊥} \subseteq E_{1}$ be the orthogonal complement of $L_{1}$ . Then $L_{1} \oplus L_{1}^{⊥} ≅ E_{1}$ as bundles over $X_{1}$ , in the convention of Whitney sum, tensor, dual, Hom, and exterior-power bundles.

Suppose $X_{j}$ , bundles $L_{1}, \dots, L_{j}$ over $X_{j}$ and a rank- $(n - j)$ bundle $L_{j}^{⊥}$ over $X_{j}$ with $q_{j}^{*} E ≅ L_{1} \oplus \dots \oplus L_{j} \oplus L_{j}^{⊥}$ have been constructed. If $j < n$ , put $X_{j + 1} : = P (L_{j}^{⊥}), π_{j + 1} : X_{j + 1} \to X_{j}, L_{j + 1} : = γ_{L_{j}^{⊥}}, q_{j + 1} : = q_{j} π_{j + 1},$ choose a metric on the numerable bundle $π_{j + 1}^{*} L_{j}^{⊥}$ and let $L_{j + 1}^{⊥}$ be the orthogonal complement of $L_{j + 1}$ in it. Iterating until $j = n$ gives the real flag bundle $Fl (E) : = X_{n}, q : = q_{n} : Fl (E) \to B,$ together with line bundles $L_{1}, \dots, L_{n}$ over $Fl (E)$ , which we keep denoting by the same symbols after pulling back along the remaining projections.

By construction, and by the pullback compatibilities of Vector-bundle pullback is canonically functorial, $q^{*} E ≅ L_{1} \oplus \dots \oplus L_{n} over Fl (E) .$ The Stiefel–Whitney roots of $E$ are the classes $t_{j} : = w_{1} (L_{j}) = x_{L_{j}} \in H^{1} (Fl (E); F_{2}), 1 \leq j \leq n,$ computed with the rank-one case of Stiefel–Whitney classes from the projective-bundle relation. For $n = 0$ we set $Fl (E) : = B$ , $q : = {id}_{B}$ , and the sum $L_{1} \oplus \dots \oplus L_{n}$ is empty, so no root is defined. For $n = 1$ the construction stops at the first stage, so $Fl (E) = P (E) ≅ B$ , $L_{1} ≅ E$ and $t_{1} = w_{1} (E)$ .

Facts & Assumptions

Given: AC, a numerable real rank- $n$ bundle $E \to B$ with $n \geq 0$ over a paracompact Hausdorff CGWH base of CW homotopy type, and the iteration above.

[F1]

For a numerable real bundle $G$ of rank $r \geq 1$ , the projective bundle $P (G) \to$ base is a numerable fiber bundle with fiber ${R P}^{r - 1}$ carrying the tautological line $γ_{G} \subseteq π^{*} G$ ; for $r = 1$ the projection is a homeomorphism and $γ_{G} ≅ G$ (Real projective bundle and tautological line).

[F2]

Under AC every numerable real or complex vector bundle admits a continuous positive-definite fiber metric (Numerable vector bundles admit bundle metrics).

[F3]

In a local frame of a topological vector bundle in which a line subbundle is spanned by the first vector, fiberwise Gram--Schmidt with a continuous bundle metric produces a continuous orthonormal frame. Hence the remaining frame vectors locally trivialize the orthogonal complement, and the addition map gives a bundle isomorphism $L \oplus L^{⊥} ≅ G$ . Whitney sums have the block-diagonal transition convention of Whitney sum, tensor, dual, Hom, and exterior-power bundles, and the relevant local-frame convention is that of Real and complex topological vector bundles.

[F4]

Under AC every vector bundle over a paracompact Hausdorff base is numerable: apply the subordinate-partition theorem to a linear chart cover (Under choice and dependent choice, every open cover of a paracompact Hausdorff space admits a locally finite subordinate partition of unity, Real and complex topological vector bundles).

[F5]

Pullback is canonically functorial, so the successive pullbacks compose and $π^{*}$ of a direct sum is the direct sum of the pullbacks (Vector-bundle pullback is canonically functorial).

[F6]

The total space of a numerable compact-Hausdorff-fiber bundle over a paracompact Hausdorff CGWH base is again paracompact Hausdorff and CGWH; its total space also has CW homotopy type when both the base and the fiber do. Every fiber used here is ${R P}^{r - 1}$ for some $r \geq 1$ , hence is a compact Hausdorff finite CW complex. Therefore every intermediate $X_{j}$ and $Fl (E)$ has all four properties (Compact-fibre bundle totals preserve paracompactness, and CW type under CW-type hypotheses).

[F7]

For a rank-one bundle $L$ the class $w_{1} (L) = x_{L}$ is defined and lies in $H^{1}$ of the base (Stiefel–Whitney classes from the projective-bundle relation).

[A1]

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

Verification

1.1

The first stage splits. By [F1] the tautological line $L_{1} = γ_{E}$ is a subbundle of $E_{1} = q_{1}^{*} E$ , and $E_{1}$ is numerable. Choose a metric by [F2]. On a local frame $(s_{1}, \dots, s_{n})$ with $s_{1}$ spanning $L_{1}$ , the Gram--Schmidt formulas divide only by the positive continuous norms of the successive nonzero orthogonalized vectors, so they produce a continuous orthonormal frame $(e_{1}, \dots, e_{n})$ with $e_{1}$ spanning $L_{1}$ . Thus $e_{2}, \dots, e_{n}$ locally frame the fiberwise orthogonal complement $L_{1}^{⊥}$ , and fiberwise addition gives $E_{1} ≅ L_{1} \oplus L_{1}^{⊥}$ as in [F3]. The complement is numerable by [F4], since $X_{1} = P (E)$ is paracompact Hausdorff by [F6].

F1F2F3F4F6A1

2.1

The iteration is legitimate and terminates. Suppose the data of stage $j$ are constructed with $q_{j}^{*} E ≅ L_{1} \oplus \dots \oplus L_{j} \oplus L_{j}^{⊥}$ and $L_{j}^{⊥}$ numerable of rank $n - j$ . If $j < n$ then $L_{j}^{⊥}$ has positive rank, so [F1] gives the numerable projective bundle $π_{j + 1} : X_{j + 1} = P (L_{j}^{⊥}) \to X_{j}$ with tautological line $L_{j + 1} = γ_{L_{j}^{⊥}} \subseteq π_{j + 1}^{*} L_{j}^{⊥}$ . Choosing a metric on $π_{j + 1}^{*} L_{j}^{⊥}$ and repeating the local Gram--Schmidt construction of step 1.1 splits $π_{j + 1}^{*} L_{j}^{⊥} ≅ L_{j + 1} \oplus L_{j + 1}^{⊥}$ with $L_{j + 1}^{⊥}$ numerable by [F4] and of rank $n - j - 1$ ; by [F5] the pulled-back splitting of $q_{j}^{*} E$ combines with this one, giving $q_{j + 1}^{*} E ≅ L_{1} \oplus \dots \oplus L_{j} \oplus L_{j + 1} \oplus L_{j + 1}^{⊥}$ . At $j = n$ no positive-rank complement remains and the iteration stops. Each $X_{j}$ is paracompact Hausdorff, CGWH, and of CW type by [F6], applied to the numerable compact-fiber bundle $π_{j}$ .

F1F2F3F4F5F6step 1.1A1

3.1

The conclusion and the degenerate cases. Substituting the terminal identity of step 2.1 gives $q^{*} E ≅ L_{1} \oplus \dots \oplus L_{n}$ over the paracompact Hausdorff CGWH CW-type space $Fl (E)$ . Each $L_{j}$ is a line bundle, so its first Stiefel–Whitney class $t_{j} = w_{1} (L_{j}) = x_{L_{j}}$ is defined by [F7] and lies in $H^{1} (Fl (E); F_{2})$ ; this is the content of [def-stiefel-whitney-classes-from-the-projective-bundle-relation]'s rank-one case. For $n = 0$ the convention gives $Fl (E) = B$ and the empty sum, and for $n = 1$ the iteration stops after step 1.1, where $X_{1} = P (E) ≅ B$ and $γ_{E} ≅ E$ , so $t_{1} = w_{1} (E)$ .

F1F7step 2.1

4.1

The fiber of the construction. Over a base point $b$ , the successive projectivizations parametrize a chain of subspaces $0 \subset L_{1} \subset L_{1} \oplus L_{2} \subset \dots \subset E_{b}$ with $\dim (L_{1} \oplus \dots \oplus L_{j}) = j$ , since each stage consists of the lines in the orthogonal complement of the previous sum. Sending such a chain to the flag $V_{j} = L_{1} \oplus \dots \oplus L_{j}$ is a bijection onto the complete flags in $E_{b}$ , with inverse obtained by taking successive orthogonal complements; the identification is compatible with the chosen metrics but its underlying set of chains does not depend on them. Hence the fiber of $Fl (E) \to B$ is the complete flag manifold of $R^{n}$ , a compact manifold, in agreement with the compactness invoked in [F6].

F1F3step 2.1step 3.1∎

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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)