指标定理（四） (9-1)

注意：指标定理（4/5）

16

Now we compute its topological index. We need to compute the Chern character of an induced bundle，and we consider this in a general setting.

Let ρ：∪ₙ → ∪ₘ be a homomorphism，and E be a complex vector bundle of rank n. Denote the induced bundle by E × ᵨ ℂᵐ. We want to find how the Chern classes of the two bundles are related.

Without loss of generality，we assume that E＝l₁⨁ · · · ⨁lₙ，that is to say，the structure group has been reduced to the torus Tⁿ. ρ defines a representation of the torus and the representation decomposes into one-dimensional ones，since Tⁿ is compact and abelian.

We first consider α：Tⁿ＝ S¹ × · · · × S¹ → S¹. Suppose it is of the form (z₁，. . .，zₙ) ↦

∏zᵢᵏⁱ，kᵢ ∈ ℤ. The S¹-bundle ET × α S¹ → BT has first chern class c(α) ∈ H²(BT；ℤ)，which is the image of the generator in H²(BS¹；ℤ). H*(BT；ℤ)＝ℤ[t₁，. . .，tₙ] is the tensor product of copies of H*(BS¹；ℤ).

Let ф：Tⁿ → Uₙ be the embedding. The bundle ET ×α ℂ → BT has transition functions ∏ fᵢᵏⁱ when ET × ф ℂⁿ＝l₁⨁ · · · ⨁lₙ → BT has transition functions (f₁，. . .，fₙ)，according to the definition of α. Thus ET × α ℂ → BT is isomorphic to l₁ᵏⁱ ⨂ · · · ⨂lₙᵏⁿ .Thus c(α)＝∑kᵢtᵢ.

In general，if α＝(α¹，. . .，αᵐ)：Tⁿ → Tᵐ，

ch(ET × α ℂᵐ)＝∑eᶜ⁽αʲ⁾＝∑eΣᵢᵏʲᵢᵗᵢ .

ⱼ ⱼ

The total Chern class is

c(ET × α ℂᵐ)＝∏(1＋c(αʲ))＝∏(1＋∑kʲᵢtᵢ).

ʲ ⱼ

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