指标定理（一）

数学使徒（MathematicalApostle）

A Brief lntroduction to lndex Theorems 指标定理

本文简要介绍的是Atiyah-Singer 指标定理及其特殊情况Hirzebruch符号差定理以及Riemann-Roch定理。粗略地说，Atiyah-Singer 指标定理陈述的是对于流形上的一类算子，通过解析的方式以及通过拓扑的方式分别得到的两种指标相等。

文中第一节是一些预备知识，包括向量丛、示性类与K理论等，如有需要可以看看或者直接跳过。第二节讲两种指标的定义、定理的叙述以及简要概括了定理证明（不是用热核那个方法）。后面是Atiyah-Singer 指标定理的特例与应用，包括de Rham复形与Dolbeault复形、Hirzebruch符号差定理以及Riemann-Roch定理的概述。

注意：文章划分（1/5）章节！

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A Brief Introduction to Index Theorems

Lan Qing

August 26, 2020

Abstract

The Atiyah-Singer index theorem states that for an elliptic operator on a com-

pact manifold, the analytical index is equal to the topological index. In this note we give a brief introduction to the Atiyah-Singer index theorem. As applications we interpret the Hirzebruch signature theorem and the Riemann-Roch theorem as special cases of the Atiyah-Singer index theorem.

Contents

1 Introduction and Preliminaries

1.1 Introduction...................................2

1.2 Fibre bundles and vector bundles.......................2

1.3 Grassmann manifolds..............................3

1.4 Principal bundles and classifying space....................5

1.5 Characteristic classes..............................

1.6 K-theory..................................... 8

1.7 Thom isomorphism in K-theory........................ 10

2 Atiyah-Singer Index Theorem 12

2.1 Analytical index.................................12

2.2 Topological index................................13

2.3 Statement of the theorem and idea of the proof............... 14

3 Examples: de Rham Complex and Dolbeault Complex 15

3.1 de Rham complex................................ 15

3.2 Dolbeault complex............................... 17

4 Hirzebruch Signature Theorem 19

4.1 Multiplicative sequence............................. 19

4.2 Hirzebruch signature theorem......................... 19

5 Riemann-Roch Theorem 22

5.1 Divisors on Riemann surfaces......................... 22

5.2 Divisors and line bundles............................23

5.3 Hirzebruch-Riemann-Roch theorem...................... 24

6 Further Developments 25

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1 Introduction and Preliminaries

1.1 Introduction

The main question in index theory is to provide index formulas for classes of Fredholm operators. Index theory has become a subject on its own only after M.F.Atiyah and I. Singer published their index theorems in a sequence of papers. Among them, Hirzebruch's signature theorem occupies a special place.Hirzebruch's theorem was generalized by A. Grothendieck, who introduced many of the ideas that proved to be fundamental for the proof of the index theorems. All these theorems turned out to be consequences of the Atiyah-Singer index theorems. ¹

In this review we give a brief introduction to some index theorems, Readers who are familiar with materials in this section can skip to the next section.

1.2 Fibre bundles and vector bundles

Let G be a topological group which acts effectively on a space F on the left.A surjection π：E→ B between topological spaces is a fiber bundle with fiber F and structure group G if B has an open cover {∪α} such that there are homeomorphisms

ф：E|∪α → ∪α × F

and the transition functions are continuous functions with values in G：

gαᵝ(x)＝фα фᵦ⁻¹|{x} × ғ ∈G

Sometimes the total space E is referred to as the fiber bundle. A fiber bundle with structure group G is also called a G-bundle. If x ∈ B，the set E᙮=π⁻¹(x) is called the fiber at x. Here the action of a group G on a space F is said to be effective if the only element of G which acts trivially on F is the identity.

A vector bundle of rank n is a fiber bundle with fber ℝⁿ and structure group GL(n，ℝ). If the fiber is ℂⁿ and the structure group is GL(n，ℂ),the vector bundle is a complex vector bundle. Unless otherwise stated,by a vector bundle we mean a C∞ real vector bundle.

Consider two vector bundles ξ and η over the same base space B. ξ is isomorphic to η. written ξ≅η，if there exists a homeomorphism f：E(ξ) → E(η) between the total spaces which maps each vector space Fb(ξ) isomorphically onto the corresponding vector space Fb(η).

Examples include the trivial bundle with total space B × Rⁿ，the tangent bundle τᴍ of a smooth manifold M，and the normal bundle ν of a smooth manifold M ⊂ Rⁿ.An important example，the Grassmann manifold，will be introduced in the next section.

A cross-section (sometimes just called a section) of a vector bundle ξ with base space B is a continuous function s : B → E(ξ) which takes each b ∈ B into the corresponding fiber Fb(ξ). Such a cross-section is nowhere zero if s(b) is a non-zero vector of Fb(ξ) for each b.

Lemma 1 An Rⁿ-bundle ξ is trivial if and only if ξ αdmits n cross-sections s₁，. . .，sₙ ωhich αre noωhere dependent.

＿＿＿＿＿＿＿＿＿＿＿＿＿

¹Index theory. Encyclopedia of Mathematics.

URL:http://www.encyclopediaofmath.org/index.php?title=Index.theory&coldid=23864

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Definition 1 A Euclidean υector bundle is α real vector bundle ξ together ωith α contin- uous function

μ：E(ξ) → ℝ

such thαt the restriction of μ to each fiber of ξ is positiυe definite αnd quadratic. The function μ itself ωill be called α Buclideαn metric on the υector bndle ξ .

A Euclidean structure on a tangent bundle is indeed a Riemannian metric.

For a vector bundle E over a manifold (which admits partition of unity)，we now show that the structure group of E may be reduced to the orthogonal group. First，we can endow E with a Riemannian structure as follows.Let {∪α} be an open cover of M which trivializes E. On each ∪α， choose a frame for E|∪α . and declare it to be orthonormal. This defines a Riemannian structure on E|∪α . Now use a partition of unity {ρα} to form a global Riemannian metric.

As trivializations of E，we take only those maps фα that send orthonormal frames of E (relative to the global metric 〈，〉) to orthonormal frames of ℝⁿ. Then the transition functions gαᵦ will preserve orthonormal frames and hence take values in the orthogonal group O(n).If the determinant of gαᵦ is positive，gαᵦ will actually be in the special orthogonal group SO(n). Similar discussion applies to the complex case.

Functorial operations on vector spaces carry over to vector bundles，For instance，if E and E' are vector bundles over M of rank n and m respectively，their direct sum E ⨁ E' is the vector bundle over M whose fiber at the point x in M is E᙮ ⨁ E'᙮· The local trivializations {фα}，{ф'α} for E and E' induce a local trivialization for E ⨁ E'：

фα ⨁ ф'α：E ⨁ E'|∪α → ∪α × (ℝⁿ ⨁ ℝᵐ).

Similarly we can define the tensor product E ⨁ E'，the dual E*.and Hom (E，E').

Let ξ be a vector bundle with projection π： E → B and let B₁ be an arbitrary topological space. Given any map f：B₁ → B one can construct the induced bundle (pullback) f*ξ over B₁ . The total space E₁ of f*ξ is the subset E₁ ⊂ B₁ × E consisting of all pairs (b，e) with f(b)＝π(e). The projection map π₁：E₁ → B₁ is defined by π₁(b，e)＝b.If (∪，h) is a local coordinate system for ξ. set U₁＝f⁻¹(∪) and define h₁：U₁ × ℝⁿ → π₁⁻¹(∪₁) by h₁(b，x)＝(b，h(f(b)，x)). Then (U₁，h₁) is clearly a local coordinate system for f*ξ . This proves that f*ξ is locally trivial.

Consider two vector bundles ξ：E(ξ) → B and η：E(η) → B over the same base space B with E(ξ) ⊂ E(η)； then ξ is a subbundle of η if each fiber Fb(ξ) is a sub-vector-space of the corresponding fiber Fb(η). Given a subbundle ξ ⊂ η， if η is provided with a Euclidean metric then a complementary summand can be constructed as follows.

Let Fb(ξ⊥) denote the subspace of Fb(η) consisting of all vectors e such that υ · ω＝0 for all ω ∈ Fb(ξ). Let E(ξ⊥)⊂ E(η) denote the union of the Fb(ξ⊥).

Lemma 2 E(ξ⊥) is the total space of α sub-bundle ξ⊥ ⊂ η. Furthermore η is isomorphic to the Whitney sum ξ ⨁ ξ⊥ .

One may construct the normal bundle of an immersion in the obvious way.

1.3 Grassmann manifolds

The Grassmann manifold Gₙ(ℝⁿ⁺ᵏ) is the set of all n-dimensional planes through the origin of the coordinate space ℝⁿ⁺ᵏ . This is to be topologized as a quotient space，as follows.

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An n-frame in ℝⁿ⁺ᵏ is an n-tuple of linearly independent vectors of ℝⁿ⁺ᵏ . The collection of all n-frames in ℝⁿ⁺ᵏ forms an open subset of the n-fold Cartesian product ℝⁿ⁺ᵏ × · · · × ℝⁿ⁺ᵏ，called the Stiefel manifold Vₙ(ℝⁿ⁺ᵏ). There is a canonical map q：Vₙ(ℝⁿ⁺ᵏ) → Gₙ (ℝⁿ⁺ᵏ) which maps each n-frame to the n-plane which it spans. Now give Gₙ(ℝⁿ⁺ᵏ) the quotient topology.

Grassmann manifolds can also be viewed as the collection of all orthogonal projections of rank n.

Lemma 3 The Grassmann manifold Gₙ(ℝⁿ⁺ᵏ) is α compαct topologicαl mαnifold of di-mension nk. The correspondence X → X⊥，ωhich αssigns to eαch n-plαne its orthogonαl k-plane，defines α homeomorphism betωeen Gₙ(ℝⁿ⁺ᵏ) αnd Gₖ (ℝⁿ⁺ᵏ).

Now we construct the tautological bundle γⁿ (ℝⁿ⁺ᵏ) over Gₙ(ℝⁿ⁺ᵏ).

Let E＝E(γⁿ(ℝⁿ⁺ᵏ)) be the set of all pairs (n-plane in ℝⁿ⁺ᵏ，vector in that n-plane). This is to be topologized as a subset of Gₙ(ℝⁿ⁺ᵏ) × ℝⁿ⁺ᵏ .The projection map π：E → Gₙ(ℝⁿ⁺ᵏ) is defined by π(X，x)＝X. One easily verifies that it is locally trivial.

Theorem 1 For αny n-plane bundle ξ over α compαct bαse spαce B there erists α bundle mαp ξ → γⁿ(ℝⁿ⁺ᵏ) proυided thαt k is sufficiently lαrge.

A bundle mαp from η to ξ is a continuous map g : E(η) → E(ξ) which carries each vector space Fb(η) isomorphically onto one of the vector spaces F∡(ξ).Setting ˉg(b)＝b'，it is clear that the resulting function ˉg：B(η) → B(ξ) is continuous.

Theorem 2 If g : E(η) → E(ξ) is α bundle mαp，αnd if ˉg : B(η) → B(ξ) is the corresponding mαp of bαse spαces，then η is isomorphic to the induced bundle ˉg*ξ.

Let ℝ∞ denote the vector space consisting of those infinite sequences x＝(x₁， x₂，x₃， . . .)with only finitely many nonzero coordinates. For fixed k，the subspace consisting of all x＝(x₁，x₂，. . .，Xₖ，0，0，. . .) will be identified with the coordinate space ℝᵏ. The infinite Grassmann manifold Gₙ＝Gₙ(ℝ∞) is the set of all n-dimensional linear sub-spaces of ℝ∞，topologized as the direct limit of the sequence Gₙ(ℝⁿ) ⊂ Gₙ(ℝⁿ⁺¹) ⊂ Gₙ(ℝⁿ⁺²) ⊂. . . . As a special case， the infinite projective space P∞＝G₁(ℝ∞) is equal to the direct limit of the sequence P¹ ⊂ P² ⊂ P³ ⊂. . . .

A canonical bundle γⁿ over Gₙ is constructed， just as in the finite dimensional case，as follows. Let E(γⁿ) ⊂ Gₙ × ℝ∞ be the set of all pairs ( n-plane in ℝ∞，vector in that n-plane )，topologized as a subset of the Cartesian product. Definen π：E(γⁿ) → Gₙ by π(X，x)＝X. In fact γⁿ is locally trivial，and we omit the proof.

Recall that a paracompact space is a Hausdorff space such that every open covering has a locally finite open refinement. The infinite Grassmann space is paracompact.

Lemma 4 For αny fiber bundle ξ oυer α pαrαcompαct spαce B，there erists α locαlly finite coυering of B by countably mαny open sets ∪₁，∪₂，∪₃，. . .，so thαt ξ|∪ᵢ is triυiαl for eαch i.

Theorem 3 Any ℝⁿ-bundle ξ oυer α pαrαcompαct bαse spαce αdmits α bundle mαp ξ → γⁿ.

Theorem 4 Any tωο bundle mαps ξ → γⁿ αre bundle-homotopic.

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Let Vectⁿℝ(B) denote the set of isomorphism classes of n-dimensional real vector bun-dles over B，[B，Gₙ] denote the set of homotopy classes of maps B → Gₙ，then we have a surjection

Vectⁿℝ(B) → [B，Gₙ]

Moreover，if B is a CW-complex (or simplicial complex)，then this correspondence is in fact one-to-one.

Theorem 5 Let ξ be α υector bundle oυer B，f₀，f₁：B' → B αre continuous mαps such thαt f₀ is homotopic to f₁.Then if B' is α CW-complex，ωe hαυe f*₀(ξ) ≅ f*₁(ξ).

The proof is done by induction on the skeleta of B'.

Therefore for a CW-complex B we have

₁：₁

Vectⁿℝ(B) ↔ [B，Gₙ(ℝ∞)]，and similar-ly

₁：₁

Vectⁿℂ(B) ↔ [(B，Gₙ(ℂ∞)]. In this sense，Gₙ is called the classifying space for n-dimensional vector bundles，γⁿ is called the universal n-plane bundle，and f：B → Gₙ is called the classifving map for ξ.

This is true for general paracompact space. See Theorem 1.6 in Vector Bundles αnd K-Theory by Hatcher. We sketch the proof in the case of a compact，Hausdorff base.

Theorem 6 Let X be compαct Hαusdorff. Let E

ᴘ

→ B be α υector bundle αnd f₀，f₁： X → B homotopic mαps. Then f*₀(E) ≅ f*₁(E).

Proof.Let h：X × l → B be a homotopy from f₀ to f₁.Then f*₀(E)＝h*(E)|x×{0} and similarly for f*₁(E).So without loss of generality we may replace B by X × l，and we wish to show that the time 0 and time 1 restrictions of a bundle E on X × l are isomorphic. Using compactness，one can show that there is a finite cover {U₁，. . .，Uₙ} of X so that the restriction of E to each Uᵢ × l is trivial.Let {φᵢ}ⁿ ᵢ₌₁ be a partition of unity subordinate to the cover {∪ᵢ}ⁿ ᵢ₋₁ . For each 0 ≤ j ≤ n，define Φⱼ＝∑ʲᵢ₌₁ φᵢ. Thus Φ₀＝0 and Φₙ = 1 on X. For simplicity，we will assume n＝2，since that is enough to see the argument. Thus we have

Φ₀＝0 ≤ Φ₁＝φ₁ ≤ Φ₂＝1

on X.For each 0 ≤ j ≤ n，we define Xⱼ ⊆ X × l to be the graph of Φⱼ.Thus X₀＝X × {0} and X₂＝X × {1}，and each Xⱼ is homeomorphic to X νia the projection. Finally，let Eⱼ be the restriction of E to Xⱼ ≅ X. We claim that E₀ ≅ E₁ ≅ E₂. To see that E₀ ≅ E₁，recall that E is trivial on ∪₁ × l. It follows that the trivialization of E on ∪₁ restricts to trivializations φ∪₁ of E₀ and E₁ on ∪₁. Define α∪₁： (E₀)|∪₁ → (E₁) |∪₁ to be the composition

(φ∪₁)|ᴇ₀ (φ∪₁)⁻¹

(E₀)|∪₁ → F × ∪₁ → (E₁)|∪₁

Now let V₁＝X–supp(φ₁). Since φ₁ is supported inside ∪₁，it follows that ∪₁∪V₁＝X. Also，we have that (E₀)|ᵥ₁＝(E₁)|ᵥ₁，since X₀ ∩(V₁ × l)＝V₁ × {0}＝X₁ ∩(V₁ × l). Now α∪₁ on (E₀)|∪₁ glues together with id on (E₀) |ᵥ₁ to give an isomorphism E₀ ≅ E₁. ▢

1.4 Principal bundles and classifying space

Let G be a Lie group. A smooth fiber bundle π：P → M with fiber G is a smooth principαl G-bundle if G acts smoothly and freely on P on the right and the fiber-preserving local trivializations

ф∪：π⁻¹(∪) → ∪ × G

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