指标定理（三） (10-7)

Here σ(P)∈ Kcₚₜ(TX)＝K(DX，∂DX)＝ˉK(Th(TX)) represents a class determined by the principal symbol of P，Ȃ (X)＝〈Â(p(TX))，[X]〉 is determined by the multiplicative sequence associated to

√x/2

────，Â(X)denotes Â(X) pulled back

sinh(√x/2)

to TX，and Td(E)＝td(c(E)) is determined by the multiplicative sequence associated to

x

────

1–e⁻ˣ

for a complex bundle E.(Here the Todd class is also pulled back to TX. See subsection 4.1 if you don’t know multiplicative sequence. )More generally，for an elliptic complex we can define the following： For any elliptic complex D＝(Dᵢ：ΓEᵢ → ΓEᵢ₊₁) over a compact oriented 2n-dimensional manifold X，

1 ₘ

indₜ(P)＝(–1)ⁿ〈(───) ∑(–1)ⁱch(Eᵢ))(Td

e(TX) ᵢ₌₀

(TX ⨂ ℂ))，[X]〉.

There’s a version of splitting principle，which says that an oriented real vector bundle of rank 2n can be pulled back over some space and splits into a direct sum of oriented plane bundles.Moreover，the mappings between cohomology groups of the base spaces are injective. Since TX is of rank 2n and is oriented，using splitting principle，we may assume that it splits into a direct sum of rank 2 oriented plane bundles，and thus it has a natural almost complex structure. After complexifying，it splits into a sum of complex line bundles l₁ ⨁ lˉ₁ ⨁ · · · lₙ ⨁ lˉₙ.Let xᵢ be the first Chern class of lᵢ. Then

xᵢ –xᵢ

Td(TX ⨂ ℂ)∏ ──── ────.

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