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Real splitting principle with mod-two injective pullback

Statement

Assume AC. Let $E \to B$ be a numerable real vector bundle of rank $n \geq 0$ over a paracompact Hausdorff CGWH base of CW homotopy type, and let $q : Fl (E) \to B$ be its real flag bundle. Then $Fl (E)$ is a base of the same kind, $q^{*} E ≅ L_{1} \oplus \dots \oplus L_{n} over Fl (E),$ and the pullback $q^{*} : H^{*} (B; F_{2}) ⟶ H^{*} (Fl (E); F_{2})$ is injective. Moreover, finitely many numerable real bundles $E_{1}, \dots, E_{m}$ over $B$ admit a common paracompact Hausdorff CGWH base of CW homotopy type $X \to B$ with $X = Fl (E_{1}) \times_{B} \dots \times_{B} Fl (E_{m})$ over which every $q^{*} E_{k}$ splits as a sum of line bundles and whose projection $X \to B$ induces an injection on $F_{2}$ -cohomology.

Facts & Assumptions

Given: AC, a numerable real rank- $n$ bundle $E \to B$ over a paracompact Hausdorff CGWH base of CW homotopy type, and its flag bundle $q : Fl (E) \to B$ .

[F1]

The flag bundle is built as the composite of the projections $X_{j} \to X_{j - 1}$ , each of which is the projective bundle of a numerable real bundle of positive rank $r_{j}$ , and $q^{*} E ≅ L_{1} \oplus \dots \oplus L_{n}$ (Real flag bundle and Stiefel–Whitney roots).

[F2]

For a numerable real rank- $r$ bundle $G$ with $r \geq 1$ over a paracompact Hausdorff CGWH base $Y$ of CW homotopy type, the projective bundle theorem gives $H^{*} (P (G); F_{2})$ free over $H^{*} (Y; F_{2})$ on $1, x_{G}, \dots, x_{G}^{r - 1}$ ; in particular the projection $P (G) \to Y$ induces an injection on $F_{2}$ -cohomology, as the inclusion of the coefficient-of- $1$ summand (Mod-two real projective bundle theorem).

[F3]

Every intermediate $X_{j}$ and $Fl (E)$ is paracompact Hausdorff CGWH of CW homotopy type, and a fiber product over $B$ of finitely many flag bundles is a numerable bundle with compact CW fiber over $B$ , hence has the same properties (Compact-fibre bundle totals preserve paracompactness, and CW type under CW-type hypotheses).

[F4]

Pullback is canonically functorial and compatible with direct sums, so the splitting of a pulled-back bundle is the pullback of the splitting (Vector-bundle pullback is canonically functorial, Whitney sum, tensor, dual, Hom, and exterior-power bundles).

[A1]

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

Proof

1.1

Every stage projection is injective on $F_{2}$ -cohomology. By [F1] the map $X_{j} \to X_{j - 1}$ is the projection of a projective bundle $P (G_{j})$ of a numerable bundle of positive rank over $X_{j - 1}$ , and $X_{j - 1}$ is admissible by [F3]; [F2] therefore makes the induced map $H^{*} (X_{j - 1}; F_{2}) \to H^{*} (X_{j}; F_{2})$ injective. The composite of finitely many injective maps is injective, so $q^{*} : H^{*} (B; F_{2}) \to H^{*} (Fl (E); F_{2})$ is injective.

F1F2F3

1.2

The splitting over the flag bundle is [F1]'s second clause, $q^{*} E ≅ L_{1} \oplus \dots \oplus L_{n}$ , with each $L_{j}$ a numerable line bundle over the admissible space $Fl (E)$ . For $n = 0$ , $Fl (E) = B$ and the sum is empty, so $q = id$ and the pullback is the identity, which is injective.

F1F3

1.3

Finitely many bundles. Let $E_{1}, \dots, E_{m}$ be numerable real bundles over $B$ , of ranks $n_{k} \geq 0$ . Define $Y_{0} = B$ and recursively $Y_{k} : = Y_{k - 1} \times_{B} Fl (E_{k}), r_{k} : Y_{k} \to Y_{k - 1} .$ We first verify the required base change. For a map $h : W \to Z$ and a positive-rank bundle $G \to Z$ , there is a fiberwise map $Ψ : P (h^{*} G) ⟶ W \times_{Z} P (G), (w, ℓ \subseteq G_{h (w)}) ⟼ (w, ℓ) .$ Over a linear chart $G ∣_{U} ≅ U \times R^{r}$ , both sides have the pulled-back projective chart $h^{- 1} (U) \times {R P}^{r - 1}$ and $Ψ$ is the identity in these coordinates. Thus $Ψ$ is a homeomorphism over $W$ , and the same coordinate description identifies $γ_{h^{*} G}$ with the pullback of $γ_{G}$ . Choose at each flag stage the pullback of the metric used over $Z$ . Fiberwise orthogonal complement then commutes with pullback, because both subbundles consist of the vectors orthogonal to the same pulled-back line. Induction through the projectivization stages of [F1] therefore gives $Fl (h^{*} E_{k}) ≅ W \times_{B} Fl (E_{k}),$ with its tautological lines identified with the pulled-back ones.

Apply this with $W = Y_{k - 1}$ and $h : Y_{k - 1} \to B$ . It identifies $r_{k} : Y_{k} \to Y_{k - 1}$ with the flag projection of $h^{*} E_{k}$ , so step 1.1 over the admissible base $Y_{k - 1}$ makes every $r_{k}^{*}$ injective. With $X = Y_{m} = Fl (E_{1}) \times_{B} \dots \times_{B} Fl (E_{m})$ and $q_{X} = r_{1} \circ \dots \circ r_{m}$ , contravariance gives $q_{X}^{*} = r_{m}^{*} \dots r_{1}^{*},$ a composite of injective maps. By [F3] every $Y_{k}$ , and in particular $X$ , is admissible. For each $k$ , let $q_{k}^{''} : X \to Fl (E_{k})$ be the $k$ -th projection. Then $q_{X} = q_{k} q_{k}^{''}$ and $q_{X}^{*} E_{k} ≅ q_{k}^{'' *} q_{k}^{*} E_{k} ≅ q_{k}^{'' *} (L_{1}^{(k)} \oplus \dots \oplus L_{n_{k}}^{(k)})$ , which by [F4] is a direct sum of the line bundles $q_{k}^{'' *} L_{j}^{(k)}$ . [F1, F3, F4, step 1.1]

2.1

Boundary cases. If some $n_{k} = 0$ then $E_{k}$ is the zero bundle, its flag bundle is $B$ and $q_{X}^{*} E_{k} = 0$ splits as an empty sum; the corresponding factor contributes no projection and does not affect injectivity. If $m = 0$ then $X = B$ , the empty family of bundles is split vacuously, and $q_{X} = id$ . If $B = \emptyset$ then all spaces are empty and the cohomology groups are zero, so injectivity is vacuous. AC is used through the metric choices of the flag construction and the projective bundle theorem, as recorded.

F1F2A1step 1.1step 1.3∎

Depends on

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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)