指标定理（四） (9-2)

By splitting principle，this is true for general rank n complex vector bundles. Here are some examples：i) ρ：∪ₙ → ∪ₙ is conjugation (of complex numbers). E × ᵨ

ℂⁿ＝E*＝ˉE. kⁱⱼ＝–δⁱⱼ.

ₙ ₙ

c(E*)＝∏(1 – xᵢ)＝∑(–1)ⁱcᵢ(E).

ᵢ₌₁ ᵢ₌₀

ii) For ∧ᵏ E，u ∈ Tⁿ acts as eᵢ₁ ∧ · · · ∧ eᵢₖ ↦ ue，∧ · · · ∧ ueᵢₖ.

c(αᵢ₁，. . .，ᵢₖ)＝xᵢ₁＋· · ·＋xᵢₖ and c(∧ᵏE)

＝∏ (1＋(xᵢ₁＋· · ·＋xᵢₖ)).

1≤i₁＜· · ·＜iₖ≤n

iii) For ∧ᵏE*，

c(∧ᵏE*)＝∏ (1–(xᵢ₁＋· · ·＋xᵢₖ)).

1≤i₁＜· · ·＜iₖ≤n

ch(∧ᵏE*)＝∑eˉ(xᵢ₁＋· · ·＋xᵢₖ)).

1≤i₁＜· · ·＜iₖ≤n

iv)

ⁿ ⁿ

∑ ch(∧ᵏE*) · tᵏ＝∏(1＋te⁻ˣⁱ).

ₖ₌₀ ᵢ₌₁

Now let’s consider an oriented real bundle E of rank 2n. By splitting principle， we may assume that it splits into a direct sum of oriented plane bundles (or complex line

17

bundles) and talk about its Chern roots. We want to calculate ∑ⁿₖ₌₀ ch(∧ᵏE* ⨂ ℂ) · tᵏ Let ρ：Tⁿ ⊂ SO₂ₙ → ∪₂ₙ be the inclusion，where Tⁿ denotes the standard maximal torus. This Tⁿ ⊂ ∪₂ₙ is a conjugate subgroup of a torus in the standard maximal torus in ∪₂ₙ， and the conjugation restricts to

(cos(2πtᵣ) –sin(2πtᵣ)

) ↦(ᵉⁱ²πᵗʳ 0)

(sin(2πtᵣ) cos(2πtᵣ) ( 0 ₑ⁻ⁱ²πᵗʳ)

on each S¹. Thus the weights of ρ：Tⁿ ⊂ SO₂ₙ → ∪₂ₙ are (x₁，–x₁，. . .，xₙ，–xₙ). where the xᵢ’s are the Chern roots. By the same argument as above，

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