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Naturality, stability, and mod-two reduction of Pontryagin classes

Statement

Assume AC. Let $E \to B$ be a numerable real bundle over a nonempty path-connected paracompact Hausdorff CW base, and let $ρ_{2}$ denote reduction mod two. Then

Naturality: $p_{i} (f^{*} E) = f^{*} p_{i} (E)$ for every continuous $f : B^{'} \to B$ with $B^{'}$ a nonempty path-connected paracompact Hausdorff CW complex;
Stability: $p_{i} (E \oplus ε^{r}) = p_{i} (E)$ for the trivial bundle $ε^{r}$ , and $p_{i} (E) = 0$ whenever $2 i > rank E$ ;
Mod-two reduction: $ρ_{2} p_{i} (E) = w_{2 i} (E)^{2}$ .

Facts & Assumptions

[A1]

The Axiom of Choice is assumed, exactly as inherited from the Chern-class and Stiefel-Whitney suppliers (The Axiom of Choice).

[F1]

$p_{i} (E) = (- 1)^{i} c_{2 i} (E_{C})$ with $p_{0} = 1$ and $p_{i} = 0$ for $2 i > rank E$ (Pontryagin classes by complexification).

[F2]

On the stated CW bases Chern classes are natural for continuous maps between such bases, multiplicative over Whitney sums, and the trivial bundle has total Chern class $1$ (Naturality, normalization, and Whitney sum for Chern classes).

[F3]

For a numerable complex bundle $V$ over a path-connected paracompact Hausdorff CW base one has $w_{2 i} (V_{R}) = ρ_{2} c_{i} (V)$ and $w_{2 i + 1} (V_{R}) = 0$ (Mod-two reduction of Chern classes).

[F4]

Total Stiefel-Whitney classes are multiplicative over Whitney sums (Whitney sum formula for Stiefel–Whitney classes).

[F5]

Whitney sums have block-diagonal transition functions, and pullback precomposes transition functions by the base map. The underlying real bundle regards complex transition functions as real-linear maps. (Whitney sum, tensor, dual, Hom, and exterior-power bundles).

[F6]

Singular cohomology over a commutative coefficient ring is graded commutative. The Stiefel–Whitney conventions are $w_{0} = 1$ , $w_{j} = 0$ above the real rank and $w (0) = 1$ . (Singular cohomology is graded commutative, Stiefel–Whitney classes from the projective-bundle relation).

Proof

technique · direct

Given: AC and a numerable real bundle $E \to B$ over the nonempty path-connected paracompact Hausdorff CW base, and a continuous $f : B^{'} \to B$ between bases of this kind.

1.1F1F2F5

Tensoring the transition functions of $E$ with $C$ before or after pulling them back gives the same cocycle, using the same real transition matrices as complex matrices for complexification, so $(f^{*} E)_{C} ≅ f^{*} (E_{C})$ . Thus [F1] and naturality of Chern classes [F2] give $p_{i} (f^{*} E) = (- 1)^{i} c_{2 i} (f^{*} (E_{C})) = (- 1)^{i} f^{*} c_{2 i} (E_{C}) = f^{*} p_{i} (E)$ .

1.2F1F2F5

The fiberwise map obtained by distributing the tensor product gives $(E \oplus ε^{r})_{C} ≅ E_{C} \oplus (ε^{r})_{C}$ and is compatible with all transition functions. The Chern class of the trivial complex bundle is $1$ by [F2], so multiplicativity gives stability; the rank cutoff is part of [F1].

1.3F3F4F5F6

For each real fiber, $v \otimes (a + i b) \mapsto (a v, b v)$ is a real-linear isomorphism from $(E_{b} \otimes_{R} C)_{R}$ to $E_{b} \oplus E_{b}$ ; its inverse is $(x, y) \mapsto x \otimes 1 + y \otimes i$ , as is checked on simple tensors and their linear combinations; both formulas are continuous in every bundle chart and compatible with real transition functions and hence defines $(E_{C})_{R} ≅ E \oplus E$ . For the complex bundle $E_{C}$ , [F3] now gives $ρ_{2} c_{2 i} (E_{C}) = w_{4 i} ((E_{C})_{R})$ , while multiplicativity [F4] gives $w ((E_{C})_{R}) = w (E)^{2}$ . By [F6], in $F_{2}$ coefficients all homogeneous classes commute (the sign becomes 1), so the cross terms in the finite square cancel in pairs and the square of a sum is the sum of squares, whose degree- $4 i$ component is $w_{2 i} (E)^{2}$ .

2.1F1step 1.2step 1.3

Combining steps 1.2 and 1.3 with the definition [F1]: $ρ_{2} p_{i} (E) = (- 1)^{i} ρ_{2} c_{2 i} (E_{C}) = (- 1)^{i} w_{2 i} (E)^{2} = w_{2 i} (E)^{2}$ , because $(- 1)^{i}$ is $\pm 1$ and the target has exponent two.

3.1A1F1F6step 1.2step 2.1∎

Boundary cases. For $i = 0$ all three assertions read $1 = 1$ ; for the zero bundle $p (0) = 1$ , $w (0) = 1$ and the identity is $1 = 1$ . The rank cutoff makes $p_{i} (E) = 0$ for $2 i > n$ while the right side $w_{2 i} (E)^{2}$ vanishes for $2 i > n$ as well, so no mismatch occurs. The empty base is excluded explicitly and $F_{2}$ is a field, so no zero-ring issue arises. AC is used only through [A1].

Source notes

Milnor-Stasheff section 15 states the naturality, stability and mod-two reduction of the Pontryagin classes; the proof above reads the mod-two identity from the Chern-class comparison $w_{2 i} = ρ_{2} c_{i}$ and the real isomorphism $(E_{C})_{R} ≅ E \oplus E$ .

Depends on

Dependency tree · 2 levels

Sources

Milnor and Stasheff, Characteristic Classes, section 15