指标定理（三）

注意：指标定理（3/5）

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The set of all equivalence classes [V₀，. . .，Vₙ；σ₁，. . .，σₙ] in 𝓛 ₙ(X。Y) will be denoted by Lₙ(X，Y). The set Lₙ(X，Y) is an abelian group under the operation ⨁. Consider the nat-ural map 𝓛 ₙ(X，Y) → 𝓛 ₙ₊₁(X，Y) which associates to each element (V₀，. . .，Vₙ；σ₁，. . .，σₙ) the trivially extended element (V₀，. . .，Vₙ，0；σ₁，. . .，σₙ，0). One verifies that for n ≥ 1 this defines an isomorphism. Consequently everything reduces to the case n＝1 which has been discussed above.

Now we have found the unique element d(π*(∧*(E))) ∈ K(E，E₀) determined by фᵢ：π*(∧ⁱE) → π*(∧ⁱ⁺¹E)，(ω，υ)∈π* (∧ⁱE) ↦ (ω，ω∧υ).

Secondly，what is the ”product” operation? Indeed K(E，E₀) is a K(M)-module.

We consider more generally. A ring homomorphism μ：K(X) ⨂ K(Y) → K(X × Y)can be defined by μ(α ⨂ b)＝p*₁(α)p*₂(b) where p₁ and p₂ are the projections of X × Y onto X and Y. We call it the externel product，and write α * b＝μ(α ⨂ b)＝p*₁(α)p*₂(b).

We would like to define a similar notion for reduced groups.

Let X∧Y＝X × Y/X ∨ Y be the smash product of X，Y. We have the long exact sequence

· · · → ˉK(S(X × Y)) → ˉK(S(X∨Y)) → ˉK(X∧Y) → ˉK(X × Y) → ˉK(X∨Y) → . . .

where the last map in the sequence is a split surjection，with splitting ˉK(X) ⨁ ∼K (Y) → ∼K(X × Y)，(α，b) ↦p*₁(α)＋p*₂(b) where p₁ and p₂ are the projections of X × Y onto X and Y. Thus the sequence is broken up into short exact sequences. In particular

0 → ˉK(X∧Y) → ˉK(X × Y) → ˉK(X∨Y) → 0

is exact.

If α ∈ ˉK(X)，b ∈ ˉK(Y)，we claim that α * b ∈ K(X × Y) indeed comes from ˉK(X∧Y). This is true since the image of α * b in ˉK(X∨Y) is 0. We define that element in ˉK(X∧Y) to be their externel product，and still denote it by α * b. This reduced externel product

ˉK(X) ⨂ ˉK(Y) → ˉK(X∧Y)

is still a ring homomorphism.

Replace the X above by X⁺ and let Y＝X/A. Here X⁺ is the pointed space(X ⊔ {*}，*). The diagonal map of X induces Δ：X/A → X⁺∧(X/A)，and consequently

Δ*

ˉK(X⁺) ⨂ ˉK(X/A) → ˉK(X⁺∧(X/A)) → ˉK(X/A).

Also ˉK(X⁺)＝K(X). The formula above makes K(X，A) into a K(X)-module，In particular， using π*，K(E，E₀) is a K(M)-module.

By the way，there's a general version of periodicity theorem. β：ˉK (X) → ˉK (S²X)＝ˉK(S²∧X)，α ↦(H – 1) * α， is an isomorphism for all compact Hausdorff spaces X.

Lastly，we talk about its proof.

If M is a point and E is the trivial line bundle， d(π*(∧*(E))) ∈ K(S²) is determined by the sequence on E₀，

0 → E × ℂ → E × E → 0，

where the mapping in the middle sends (ω，υ) to (ω，ω∧υ)＝(ω，υ). It suffices to show that this is a generator of K(S²)＝ℤ. Indeed this is just the generator 1 – H.

The proof uses K-theory with compact support. If X is a locally compact space，and X∞ is its one point compactification，define Kcₚₜ(X)＝ˉK(X∞)，K⁻ⁱcₚₜ(X)＝Kcₚₜ(X × ℝⁱ).

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Elements in Kcₚₜ(X) are represented by formal difference of vector bundles trivialized outside a compact set. By definition Kcₚₜ(E) ＝ˉK(Th(E)). By the discussion above，Kˉ*cₚₜ (E)is a Kˉ*cₚₜ(M)-module.

We say that a class u ∈ Kcₚₜ(E) is a K-theory orientation of E if Kˉ*cₚₜ(E) is a free Kˉ*cₚₜ(M)-module with generator u.A class u ∈ Kcₚₜ(E) is said to have the Bott periodicity property if u determines a K-theory orientation in any local trivialization of E over a closed subset C ⊂ M.

One can verify that，as the trivial case above，d(π*(∧*(E))) has the Bott periodicity property.

We claim that if the base space is compact and u ∈ Kcₚₜ(E)has the Bott periodicity property，then u ∈ Kcₚₜ(E) is a K-theory orientation of E.Pick a covering of M by finitely many closed subsets such that E is trivial when restricted to any of them. For the theory Kˉ*cₚₜ，there exists Mayer-Vietories sequence as well. ∪sing Mayer-Vietories sequence and five lemma, we construct the desired isomorphism for A∪B from the known isomorphisms for A，B，A∩B. Using induction，the proof is completed.

In particular，the mapping

ψ：K(M)＝Kcₚₜ (M) → K (E，E₀)＝ˉK(Th(E))＝Kcₚₜ(E)，

α ↦ π*α · d(π*(∧*(E)))

is an isomorphism.

2 Atiyah-Singer Index Theorem

2.1 Analytical index

Let X be a compact，differentiable manifold，and E，F be complex vector bundles over X. Let Γ(E) denote the space of smooth sections of E. A complex linear operator D：Γ(E) → Γ(F)，locally can be viewed as a mapping from smooth vector-valued functions on a Euclidean space to another such space. We say that D is a differential operator if locally，it can be written as

∂|α|

D＝∑ Aα(x)──

|α|≤k ∂xα

where α is a multiindex and Aα is a matrix-valued function defined locally on X. As one can verify，changing trivializations of the bundles leads to a similar formula.The greatest degree appearing in the formula above is independent of charts and trivializations, and is defined to be its order.

We define the principal symbol of a differential operator of order m as follows.{iᵐAα}|α|₌ₘ represents a well-defined section of (⨀ᵐTX) ⨂ Hom(E，F)，where ⨀ᵐTX denotes sym-metric tensor product and locally has basis ∂|α|

───，|α|＝m.

∂xα

The section σ is defined to be the principal symbol. Given a covector ξ＝ξᵢ dxⁱ at x，we have σξ＝iᵐ ∑|α|₌ₘ Aα(x)ξα：E᙮ → F᙮. We mention here that some authors might omit iᵐ，which indeed does not matter.

Definition 2 D is elliptic if ∀x，∀ξ ≠ 0 αt x，σξ is αn isomorphism. More generαlly，α chαin complex Dᵢ：Γ(Eᵢ) → Γ(Eᵢ₊₁) is elliptic if ∀x，∀ξ ≠ 0 αt x，its principαl symbols induce αn erαct sequence of fibers αt x.

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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.

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

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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]〉.

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)∏ ──── ────.

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).

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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)

indₜ(M × N，V ⨂ W)＝indₜ(M，V) · indₜ(N，W). We construct a cobordism ring Ω(B∪) as follows. We say that (M，V) ~ (N，W)， if there is a manifold K such that ∂K＝M ⊔ N， and there’s a complex vector bundle P restricting to V and W on M and N respectively. iv) [M，V]＝0 in Ω(B∪) implies indₜ(M，V)＝0. Ω(B∪) ⨂ ℚ is a polynomial ring generated by [ℂP²ⁿ，1] and [S²ⁿ，αₙ] where αₙ ∈ ˉK(S²ⁿ)＝ℤ is a generator. We find that

indₜ[ℂP²ⁿ，1]＝1，indₜ[S²ⁿ，αₙ]＝2ⁿ.

Now intₜ is completely determined. In order to finish the proof，it suffices to show that the same properties hold for intα.

Here comes a problem. In order to define intα on K(T*M，T*M₀)， we need to show that every class can be represented by a differential operator. But this is not true in general. However, this is true for pseudodifferential operators.

Note that this is a generalization of the proof of the Hirzebruch index theorem，which we will talk about later.

3 Examples:de Rham Complex and Dolbeault Com-plex

3.1 de Rham complex

Let X be a compact，differentiable manifold of dimension k and T its tangent bun-dle. Let Eᵢ＝∧ⁱ (T* ⨂ ℂ). The exterior derivative d yields an elliptic complex，d＝(d：ΓEᵢ → ΓEᵢ₊₁). Locally ∂f

d：fdxⁱ¹∧ · · · dxⁱᵏ ↦df∧dxⁱ¹∧ · · · dxⁱᵏ

∂f

＝── dxʲ∧dxⁱ¹∧ · · · dxⁱᵏ， ∂xʲ

where the Einstein convention is used. Thus it is of order one，and

σ(x，υ)＝iυⱼdxʲ∧dxⁱ¹ ∧ · · · dxⁱᵏ＝iυ ∧ dxⁱ¹ ∧ · · · dxⁱᵏ.

Its symbol σ(x，υ) is the wedge-product ω ↦ iυ ∧ ω (as a linear map). The cohomology groups Hⁱ will also be denoted by Hⁱdeʀhαm · By the theorem of de Rham，we have

HⁱdeRhαm＝Hⁱ(X；ℂ).

indα(d)＝∑(–1)ⁱdim(HⁱdeRhαm)＝e(X).

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