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First Pontryagin class of an underlying real complex bundle

Statement

Assume the Axiom of Choice. Let $B$ be a nonempty path-connected paracompact Hausdorff compactly generated weak Hausdorff space of CW homotopy type, and let $E \to B$ be a numerable complex vector bundle of finite rank. These are the extended-base hypotheses in the integral complex projective bundle theorem used to define the characteristic classes here. Write $E_{R}$ for its underlying real bundle. With the normalization $p_{1} (V) = - c_{2} (V \otimes_{R} C)$ for a real bundle $V$ , one has $p_{1} (E_{R}) = c_{1} (E)^{2} - 2 c_{2} (E) in H^{4} (B; Z) .$

Facts & Assumptions

Given: The stated bundle over $B$ and integral Chern and Pontryagin classes with the supplier conventions.

[F1]

Pontryagin classes by complexification defines $p_{i} (V) = (- 1)^{i} c_{2 i} (V \otimes_{R} C)$ for a real bundle over a path-connected CW-type base; in particular $p_{1} (V) = - c_{2} (V \otimes_{R} C)$ .

[F2]

Complexification is conjugation invariant gives $c_{i} (\overline{E}) = (- 1)^{i} c_{i} (E)$ for numerable complex bundles under AC.

[F3]

Naturality, normalization, and Whitney sum for Chern classes gives naturality and $c (E \oplus F) = c (E) c (F)$ for numerable complex bundles over a path-connected CW base.

[A1]

Full AC is assumed as in The Axiom of Choice.

[F4]

Integral complex projective bundle theorem extends the projective-bundle basis and unique monic relation to paracompact Hausdorff compactly generated weak Hausdorff spaces of CW homotopy type, the exact base class imposed in the Statement. Pullback of the projective bundle and its tautological line pulls this relation back; uniqueness makes the Chern coefficients natural for the CW-model map.

[F5]

Homotopic maps induce equal maps in singular cohomology states that homotopic maps induce equal pullbacks on singular cohomology. Since the map $f$ in step 1.1 is a homotopy equivalence, its pullback $f^{*}$ is injective.

Proof

technique · Direct calculation from the defining characteristic classes

1.1A1F1F3F4F5given

If $E$ has rank zero, then $E_{R}$ is the zero bundle and the zero-rank conventions in [F1] and [F3] make both sides vanish. Assume henceforth that $E$ has positive rank. Choose a homotopy equivalence $f : C \to B$ from a nonempty path-connected CW complex. Pull the numeration of $E$ back to $f^{*} E$ . The projective-bundle relation and its tautological class pull back, so uniqueness in [F4] gives $c_{i} (f^{*} E) = f^{*} c_{i} (E)$ ; the same argument applies to complexification and its Chern classes. Also $f^{*}$ is injective by [F5]. Thus it suffices to prove the identity on $C$ , where [F2] and [F3] apply. Work on $C$ for the characteristic-class calculation below.

2.1step 1.1givenalgebra

The complexification $(E_{R})_{C}$ has the complex-linear endomorphism induced by the complex structure $J$ on $E_{R}$ . Its $+ i$ and $- i$ eigensubbundles are respectively $E$ and its conjugate bundle $\overline{E}$ : the projections $(1 - i J) / 2$ and $(1 + i J) / 2$ are continuous and commute with all complexified transition functions. Hence $(E_{R})_{C} ≅ E \oplus \overline{E}$ .

3.1F2F3step 2.1algebra

Put $a = c_{1} (E)$ and $b = c_{2} (E)$ . By [F2], $c_{1} (\overline{E}) = - a$ and $c_{2} (\overline{E}) = b$ . By [F3] and step 2.1, the degree-four part of $c (E) c (\overline{E})$ is $b + a (- a) + b = 2 b - a^{2}$ , so $c_{2} ((E_{R})_{C}) = 2 c_{2} (E) - c_{1} (E)^{2}$ .

4.1F1F5step 1.1step 3.1algebra∎

Apply [F1] to $E_{R}$ and negate the equality in step 3.1. This gives $p_{1} (E_{R}) = c_{1} (E)^{2} - 2 c_{2} (E)$ . Naturality and injectivity of $f^{*}$ from step 1.1 descend this equality to the original base $B$ ; the rank-zero case was discharged separately in step 1.1.

Statement references

Pontryagin classes by complexification; Complexification is conjugation invariant; Naturality, normalization, and Whitney sum for Chern classes.

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Vector Bundles & K-Theory, section 3.2