指标定理（三） (10-8)

ᵢ 1 – e⁻ˣⁱ 1 – eˣⁱ

Noting that the Euler class of the tangent bundle is ∏ xᵢ，we arrive at the following formula of topological index：

ₘ

indₜ(D)＝〈((∑ (–1)ⁱ · ch (Eᵢ) )

ᵢ₌₀

ₙ xⱼ 1

∏ (─── · ───))，[X]〉.

ⱼ₌₁ 1 – e⁻ˣʲ 1 – e⁻ˣʲ

2.3 Statement of the theorem and idea of the proof

Theorem 10 Let X be α compαct oriented 2n-dimensionαl mαnifold αnd D αn elliptic complex oυer X. Then

indα (D)＝indₜ(D).

15

There are various ways to prove the theorem，including methods using heat kernel. Here we only present the idea of one of the methods, after which we will focus on important special cases of the theorem.

We consider the case of an elliptical operator. Write [σ(D)] for [π*(E)，π*(F)，σ(D)] ∈ K (T*M，T*M₀).[u] ↦〈ф⁻¹ ch [u]Td(TM ⨂ ℂ)， [M]〉is a function defined on K(T*M，T*M₀)， which we still denote by indₜ.Since K(T*M，T*M₀) is a K(M)-module，we define

indₜ((M，V)＝indₜ(V · [σ(D)])，

where V ∈ K(M). This function has the following properties： i) indₜ(M ⊔ N，V ⊔ W)＝ indₜ(M，V)＋indₜ(N，W). ii) indₜ(M，V ⨁ W)＝indₜ(M，V)＋indₜ(M，W). iii)

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