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Odd Chern classes of a complexified real bundle are two-torsion

Statement

Assume AC. Let $E \to B$ be a numerable real vector bundle over a path-connected paracompact Hausdorff CW base and let $E_{C}$ be its complexification. Then for every $j \geq 0$ $2 c_{2 j + 1} (E_{C}) = 0 in H^{4 j + 2} (B; Z) .$

No integral vanishing of $c_{2 j + 1} (E_{C})$ is asserted.

Facts & Assumptions

[A1]

The Axiom of Choice is assumed, inherited from the conjugation-invariance supplier (The Axiom of Choice).

[F1]

On a path-connected paracompact Hausdorff CW complex, every numerable complex bundle $V$ satisfies $c_{i} (\overline{V}) = (- 1)^{i} c_{i} (V)$ ; for a numerable real bundle $E$ on that base, the complexification $E_{C}$ is canonically isomorphic to its conjugate (Complexification is conjugation invariant).

Proof

technique · direct

Given: AC, a numerable real bundle $E \to B$ over a path-connected paracompact Hausdorff CW complex, its complexification $E_{C}$ , and an index $j \geq 0$ .

1.1F1given

By [F1] the bundle $E_{C}$ is isomorphic to $\overline{E_{C}}$ , and conjugation acts on its Chern classes by $c_{i} \mapsto (- 1)^{i} c_{i}$ .

2.1F1step 1.1

Applying the conjugation formula in odd degree $i = 2 j + 1$ to $V = E_{C}$ gives $c_{2 j + 1} (E_{C}) = (- 1)^{2 j + 1} c_{2 j + 1} (E_{C}) = - c_{2 j + 1} (E_{C})$ , using the isomorphism of step 1.1.

3.1step 2.1

Moving the right-hand side to the left gives $2 c_{2 j + 1} (E_{C}) = 0$ in the abelian group $H^{4 j + 2} (B; Z)$ , which is the assertion; the argument shows no integral vanishing, since a two-torsion class need not be zero.

4.1A1F1step 3.1∎

Boundary cases. For $j = 0$ the identity reads $2 c_{1} (E_{C}) = 0$ . For the zero bundle both sides vanish; the empty base is excluded by the path-connected hypothesis and the coefficient group is $Z$ . Division by $2$ is never performed, so the conclusion is valid integrally and no localization hypothesis is hidden.

Source notes

Miller's Lecture 36 records that the odd Chern classes of a complexified real bundle are two-torsion; the corollary above is the elementary consequence of the conjugation symmetry, and the companion counterexample on the examples page shows that the classes need not vanish integrally.

Depends on

Dependency tree · 2 levels

Sources

Miller, MIT 18.906 Algebraic Topology II, Lecture 36