指标定理（三） (10-5)

We mention without proof that for an elliptic complex，all cohomology groups are finite dimentional as complex vector spaces.

Definition 3 The (αnαlyticαl) index of αn elliptic complex

0 → Γ(E₀) → · · · → Γ(Eₘ) → 0

is defined to be

ₘ

indα(D)＝∑(–1)ᵐdimℂ(Hʲ(D)).

ⱼ₌₀

In pαrticulαr the index of αn elliptic operαtor D is dimℂ(ker(D)) – dimℂ(coker(D)).

Examples of elliptic operators include the Laplace-Beltrami operator defined on s-mooth sections of the trivial line bundle，whose principal symbol σξ is multiplication by –||ξ||².We will see more examples later.

2.2 Topological index

Let π：T*X → X be the projection，and T*X₀ denote the cotangent bundle with zero section deleted.

For an elliptic operator D：Γ(E) → Γ(F)，its principal symbol may be viewed as a map from π*(E)to π*(F). It restricts to an isomorphism outside the zero section，by the def-inition of elliptic operator. Thus it defines a class [π*(E)，π*(F)，σ(D)] ∈ K (T*X，T*X₀).

Suppose X is oriented. Recall that the Chern character gives a group homomorphism

ch：K(X) → HΠ(X；ℚ)，

and also

ch：K(X，A)＝ˉK(X/A) → HΠ(X，A；Q，ℚ)＝ˉHΠ(X/A；ℚ).

We fix the coefficient ring ℚ and write H*(T*X，T*X₀) for HΠ(T*X，T*X₀；ℚ). Thus we have

ch：K(T*X，T*X₀) → H*(T*X，T*X₀)，

and Thom isomorphism

ф：H*(X) → H*(T*X，T*X₀).

Consequently 〈ф⁻¹ch[π*(E)，π*(F)，σ(D)]，[X]〉Xis a number. But this is not what we want.

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