指标定理（三） (10-6)

Recall Thom isomorphism theorem in K-theory，saying that given a complex vec-tor bundle E → M，we have an isomorphism ψ：K(M) → K(E，E₀)，α ↦ π* (α) ·

d(π*(∧*(E))). The following diagram is not commutative：

ch

K(T*M，T*M₀) → H*(T*M，T*M₀)

ψ↑ ch ф↑

K(M) → H*(M)

Also，the element μ(E)：＝ф⁻¹ chψ[1] is a characteristic class，and hence can be represented by Chern classes. Indeed，μ＝∏ᵢ

1–eωⁱ

(───)

ωᵢ

，where the ωᵢ’s are the Chern roots. Its inverse

ωᵢ

∏ᵢ(───)

1–eωⁱ

is defined to be the Todd class. But some people define Todd class of the

14

complex vector bundle E to be Td(E)＝

ωᵢ

∏ᵢ(───)

1–e⁻ωⁱ

，i.e. the multiplicative sequence associated

to the series

x

td(x)＝───.

1–e⁻ˣ

We use the latter convention. In fact，either choice will lead to the same formula in the Atiyah-Singer index theorem. Now we arrive at the definition. Note that the tangent bundle may be identified with cotangent bundle， using a Riemannian metric.

Definition 4 The topologicαl index of αn elliptic operαtor D is

indₜ(D)＝〈ф⁻¹ch[π*(E)，π*(F)，σ(D)]Td(TM ⨂ ℂ)，[M]〉.

We mention some equivalent forms of the topological index. For any elliptic operator P over a compact oriented n-dimensional manifold X，

indₜ(P)＝(–1)ⁿ〈ch(σ(P))(Aˉ(X)²)，[TX]〉＝ (–1)ⁿ〈ch(σ(P))(Td(TX ⨂ ℂ))，[TX]〉.

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