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Realification of a complex line compares c-one, w-two, and Euler

Example

Assume AC. Let $L \to B$ be a numerable complex line over a path-connected CW base, and let $ρ_{2}$ denote reduction mod two. Then $e (L_{R}) = c_{1} (L), w_{1} (L_{R}) = 0, w_{2} (L_{R}) = ρ_{2} c_{1} (L) .$

Facts & Assumptions

Given: AC and a numerable complex line $L$ over a path-connected CW base.

[A1]

The Axiom of Choice is assumed, exactly as inherited from the characteristic-class suppliers (The Axiom of Choice).

[F1]

For a complex rank- $n$ bundle one has $c_{n} (E) = e (E_{R})$ in the complex orientation (Top Chern class equals Euler class of the underlying real bundle).

[F2]

For a complex bundle $w_{2 i + 1} (E_{R}) = 0$ and $w_{2 i} (E_{R}) = ρ_{2} c_{i} (E)$ (Mod-two reduction of Chern classes).

[F3]

An orientable real bundle has $w_{1} = 0$ (The first Stiefel–Whitney class classifies orientability).

[F4]

The underlying real bundle of a complex line carries the complex orientation (The complex orientation of the underlying real bundle).

[F5]

$c_{0} = 1$ , $c_{1} = e$ on a line, and $c_{i} = 0$ for $i \geq 2$ on a line (Chern classes from the projective-bundle relation).

Verification

technique · direct

1.1

The Euler comparison: [F4] supplies the complex orientation of $L_{R}$ , so [F1] with $n = 1$ gives $e (L_{R}) = c_{1} (L)$ .

F1F4

1.2

Orientation: $L_{R}$ is oriented by [F4], hence orientable, so $w_{1} (L_{R}) = 0$ by [F3].

F3F4

1.3

The mod-two comparison: [F2] with $i = 1$ gives $w_{2} (L_{R}) = ρ_{2} c_{1} (L)$ and $w_{3} (L_{R}) = 0$ ; by [F5] all higher Chern classes of $L$ vanish, so $w_{k} (L_{R}) = 0$ for $k \geq 3$ as well.

F2F5

2.1

Boundary cases. The trivial line $L = ε^{1}$ has $c_{1} = 0$ , $e = 0$ and $w_{1} = w_{2} = 0$ ; the restriction to rank one is the exact range in which [F5] applies, and the general-rank versions are the cited theorems. The coefficient field $F_{2}$ is nonzero, so $ρ_{2}$ is the standard reduction. AC is used only through [A1].

A1F1F2step 1.1∎

Source notes

The comparison $e (L_{R}) = c_{1} (L)$ , $w_{1} = 0$ , $w_{2} = ρ_{2} c_{1}$ for a complex line is the rank-one case of Milnor-Stasheff sections 14-15; it is used on the companion page to identify the two-torsion class of the complexified universal real line.

Depends on

Used by

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Dependency tree · two levels

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Sources

Milnor and Stasheff, Characteristic Classes, sections 14-15 (standard reference, not scraped)