指标定理（二）

注意：指标定理（2/5）

6

are G-equivariant, where G acts on U × G on the right by

(x，h) · g＝(x，hg).

For a vector bundle η : E → M，let Fr(E)＝⊔Fr(Eₓ) be the collection of all frames in the fibers of E，and π：Fr(E) → M be the map sending Fr(Eₓ) to x. Fr(Eₓ) can be identified with GL(n，ℝ)，and GL(n，ℝ) acts on Fr(E) on the right. This makes it into a principal GL(n，ℝ)-bundle.

The transition maps of π：Fr(E) → M are the same as those of η： E → M. Thus from a principal GL(n，ℝ)-bundle，we can recover the associated vector bundle using those transition maps.

Let π：P → M be a principal G-bundle and ρ： G → GL(V) a representation of G on a finite-dimensional vector space V. We write ρ(g)υ as g · υ or even gυ.The associated bundle E：＝ P × ᵨ V is the quotient of P × V by the equivalence relation

(p，υ) ~ (pg，g⁻¹ · υ) for g ∈ G and (p，υ) ∈ P × V

We denote the equivalence class of (p，υ) by [p，υ]. The associated bundle comes with a natural projection β：P × ᵨ V → M，β (l，[p，υ])＝π(p).

Theorem 7 If EG → BG is α principαl G-bundle such thαt EG is homotopicαllu triviαl，then for αn αrbitrαry principαl G-bundle π：E → B oυer α simpliciαl complex B (or more generαlly，α CW-complez)，there’s α mαp f：B → BG such thαt f*EG＝E. Equiυαlently，there’s α homomorphism F：E → EG.

Proof. We may assume that E is trivial when restricted to each simplex. We use induction. The result is clear on 0-skeleton. Suppose now we have F|ᴇᵏ：Eᵏ＝π⁻¹(Bᵏ) → EG. For a (k ＋1)-simplex σ of B，F|π⁻¹(∂σ) is determined by F|∂σ×e：∂σ × e → E. Since EG is homotopically trivial，F|∂σ×e can be extended to F|σ×e，and then to F|π⁻¹(σ) by G-equivariance. ▢

Such a bundle is called the universal G-bundle，and BG is called the classifying space of G.

Also，the map defined above is determined up to homotopy. Given two homomor-phisms F₀，F₁：E → EG，consider the bundle l × E → I × G. F₀，F₁ gives a map ∂l × E → EG，so we can extend the map to l × E → EG as above.

Write P(B，G) for the collection of (isomorphism classes of) principal G-bundles over B. By the discussion above we have a surjective mapping

P(B，G) → [B，BG].

Indeed it’s bijective，and this is why BG is called the classifying space. Bijectivity is proved using covering homotopy property.

If there is another universal G-bundle E'G → B'G，there are induced maps f：BG → B'G，g：B'G → BG such that fg，gf are homotopic to identity. Thus the classifying space，if exists，is determined up to homotopy equivalence. In particular，H*(BG；R) is completely determined.

We mention some important examples. The universal ℤ₂-bundle is S∞ → ℝP∞. The universal S¹-bundle is S∞ → ℂP∞，induced from S²ⁿ⁻¹ → ℂPⁿ.The Grassmann mani-fold Gₙ (ℝⁿ⁺ᵏ) is a quotient of Vₙ (ℝⁿ⁺ᵏ). Letting k → ∞，we get the universal GL(n，ℝ)-bundle Vₙ(ℝ∞) → Gₙ(ℝ∞). Replacing Vₙ(ℝⁿ⁺ᵏ) by the subset of all orthogonal frames

V⁰ₙ(ℝⁿ⁺ᵏ)＝O(n＋k)/O(k)，we get the universal O(n)-bundle V⁰ₙ(ℝ∞) → Gₙ(ℝ∞).

7

If H ⊂ G is a subgroup. EG → BG is the universal G-bundle，then we have the universal H-bundle EG → EG/H. Since every compact Lie group can be embedded into

O(k) for some k，for compact Lie groups we know the existance of universal bundles.

1.5 Characteristic classes

We may identify a real vector bundle with its associated principal GLₙ(ℝ)-bundle, or Oₙ-bundle if a metric is provided，or SOₙ-bundle if it’s additionally orientable，and similarly for complex case.

First we define the Chern class. We have that

H*(ℂP∞；ℤ)＝H*(BT¹：ℤ)＝ℤ[t].

Thus

H*(BTⁿ；ℤ)＝ℤ[t₁，. . .，tₙ].

For the inclusion ρ： BTⁿ → B∪ₙ，we have a homomorphism

ρ*：H*(B∪ₙ：ℤ) → H*(BTⁿ；ℤ).

This is indeed an injection.We can show that its image is invariant under Weyl group，and thus is contained in the subalgebra of symmetric polynomials，which is generated by elementary symmetric polynomials σ₁，. . .，σₙ. Calculating the Poincaré series， we find out that on each degree，H*(B∪ₙ；ℤ) and ℤ[σ₁，. . .，σₙ has the same rank as abelian groups. However，replacing ℤ with arbitrary coefficient field R，we still have an injection ρ*：H*(B∪ₙ：R) → H*(BTⁿ；R)， which implies that

H*(B∪ₙ；ℤ)＝ℤ[σ₁，. . .，σₙ] ⊂ H*(BTⁿ；ℤ)＝ℤ[t₁，. . .，tₙ]，

where σ₁，. . .，σₙ are the elementary symmetric polynomials of t₁，. . .，tₙ.

We define σ₁，. . .，σₙ ∈ H*(B∪ₙ；ℤ) to be the Chern classes of the universal rank n complex vector bundle.

For general complex vector bundle π：E → B (with metric)，by the discussion above，there exists a map f：B → B∪ₙ which pulls back the universal bundle to get E and is determined up to homotopy. For each k，f*σₖ ∈ H*(B；ℤ) is well-defined and we define it to be the k-th Chern clαss of E，denoted by cₖ(E).c(E)：＝1＋c₁(E)＋c₂(E)＋ · · · ∈H*(B；ℤ) is the total Chern class.

Similarly，for the real case we have an injection of algebras

ρ*：H*(BOₙ；ℤ₂)＝ℤ₂[ω₁，. . .，ωₙ] → H* (Bℤⁿ₂；ℤ₂)＝ℤ₂[t₁，. . .，tₙ]

where ω₁，. . .，ωₙ are elementary symmetric polynomials. These ω₁，. . .，ωₙ are called Stiefel-Whitney clαsses.

For a real vector bundle ξ of rank n，we define the i-th Pontrjagin class to be pᵢ(ξ)＝

(–1)ⁱc₂ᵢ(ξ ⨂ ℂ)∈ H⁴ⁱ(B；ℤ). Indeed，H*(BOₙ；ℚ) is the polynomial ring generated by the universal Pontrjagin classes p₁，. . .，pₘ where m＝[n/2]. We sketch a proof. Let T ⊂ Oₙ be the maximal torus in Oₙ. ℝⁿ decomposes into 2-dimensional subspaces Uⱼ，j＝1，. . .，m (and possibly one additional 1-dimensional subspace). After complexifying，each Uⱼ ⨂ ℂ decomposes into Vⱼ ⨁ ˉVⱼ . The total Chern class of the complexified bundle is ∏ⱼ(1–x²ⱼ)，so the total Pontrjagin class is p＝∏ⱼ(1＋x²ⱼ)，i.e.

ρ*：H*(BOₙ：ℚ) → H*(BT；ℚ)，p↦∏(1+x²ⱼ).

ⱼ

8

It follows that H*(BOₙ：ℚ) is the subalgebra of symmetric polynomials of x²₁，. . .，x²ₘ， which gives the conclusion.

Similarly we can calculate H*(BSOₙ；ℚ)， since the maximal torus in Oₙ is indeed in SOₙ. If n＝2m＋1 is odd，H*(BSOₙ；ℚ) is still the subalgebra of symmetric poly-nomials of x²₁，. . .，x²ₘ，i.e. the polynomial ring generated by the universal Pontrjagin classes p₁，. . .，Pₘ. But if n＝2m is even，H*(BSOₙ：ℚ) is the subalgebra of symmetric polynomials generated by x²₁，. . .，x²ₘ₋₁，x₁x₂. . .xₘ. Indeed

H*(BSO₂ₘ：ℚ)＝ℚ[p₁，. . .pₘ₋₁，χ]

where χ is the Euler class defined for oriented real bundles.

Let HΠ(B；ℤ) denote the collection of formal power series. Consider the expression

ₙ

∑ eᵗⁱ ∈ HΠ(BTⁿ；ℚ).

ᵢ₌₁

It’s symmetric so it belongs to HΠ(B∪ₙ；ℚ). We define its pullback in HΠ(B；ℚ) to be the Chern character of π：E → B，denoted by ch(E). Also the pullback of

tᵢ

∏ ─── ∈ HΠ(BUₙ；ℚ) ⊂ HΠ(BTⁿ；ℚ)

ᵢ 1 – e⁻ᵗⁱ

is defined to be the Todd clαss.

Of course，there are other ways to introduce characteristic classes，but we are not going to discuss here.

1.6 K-theory

We are assuming the connectedness of the base space so that the dimension of a vector bundle is well-defined. Let V(X) be the set of isomorphism classes of vector bundles over X. This is a semigroup under direct sum. We say that two pairs of vector bundles (E，F)，(E'，F') are equivalent if ∃A ∈ V(X)，E ⨁ F' ⨁ A = E' ⨁ F ⨁ A. The set K(X) of equivalence classes is now a group，called the K-group. V(X) is clearly identified with a subset of K(X).

K(X) is functorial. Given f：X → Y，we have pullback f*：K(Y) → K(X). If (X，*)is a pointed space，f：* → X induces a surjection f*：K(X) → K(*)＝ℤ，whose kernel is defined to be the reduced K-group ˉK(X). Both reduced and unreduced K-groups are rings with multiplication given by the tensor product.

The sequence 0 → ˉK(X) → K(X) → ℤ → 0 splits，so K(X)＝ˉK(X) ⨁ ℤ.

Lemma 5 If bαse spαce X is compαct，then ∀ υector bundle E，∃F such thαt E ⨁ F is triυiαl.

Consider the map α：V(X) → ˉK(X) ⊂ K(X)，E ↦ [E，dimE]， where we use a mumber to denote a trivial bundle. Then α(E)＝α(F) if and only if ∃ trivial bundles k，l such that E ⨁ k＝ F ⨁ l. We call two bundles stαbly equiυαlent if the above condition is satisfied. Also α is surjective since [E，F]＝[E ⨁ A，n] for some A by the lemma. This gives another characterization of ˉK(X).

These statements are true for spaces having the homotopy type of a compact space.

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Now we talk about complex bundles. The inclusion Uₙ ⊂ Uₙ₊₁ induces jₙ： BUₙ → BUₙ₊₁.If f is the classifying map for a rank n bundle E，jₙ◦f is the classifying map for E ⨁ 1. E and E ⨁ 1 are stably equivalent. Also jₙ induces (jₙ)*：[X，B∪ₙ] → [X，B∪ₙ₊₁]，and we write [X，B∪] for the direct limit of the direct system consisting of {[X，B∪ₙ]}. Thus we have

[X，B∪]＝ˉK(X).

Theorem 8 ˉK(Sⁿ) is isomorphic to ℤ if n is eυen，αnd 0 ifn is odd.

A generator of ⁻K(S²) is given by H – 1 where H is the hyperplane bundle. The first Chern class sends H to 1∈ ℤ，and hence defines an isomorphism ˉK(S²)＝ℤ.

Define K(X，A)＝ˉK(X/A).where the pair (X，A) is assumed to satisfy the homotopy extension property，e.g. a CW-pair. Then we have exact sequences

K(X，A) → ˉK(X) → ˉK(A)，

K(X，A) → K(X) → K(A).

We use SX，CX to denote the reduced suspension of X and the reduced cone over X respectively. Then we have exact sequences

ˉK(SA) → ˉK(X∪CA) → K(X)，

· · · → ˉK(S(X/A)) → ˉK(SX) → ˉK(SA) → ˉK(X/A) → ˉK(X) → ˉK(A).

Define ˉK⁻ⁿ(X)＝ˉK(SⁿX). Periodicity theorem tells us that ˉK(S²X)＝ˉK(X) holds for arbitrary space X，and hence we can also extend the ”Puppe sequence” to the right.

Also there is an unreduced version

· · · → Kⁿ(X，A) → Kⁿ (X) → Kⁿ (A) → Kⁿ⁺¹ (X，A) → Kⁿ⁺¹(X) → Kⁿ⁺¹ (A) → . . . .

K-theory is a generalized cohomology theory.

We now give another characterization of K(X， Y). Consider the collection 𝓛 (X，Y) of all triples A＝(E，F，σ) where E，F are complex vector bundles over X and σ：E|ʏ → F|ʏ is an isomorphism. Two triples (E，F，σ)， (E'，F'，σ') are said to be isomorphic if there exists bundle isomorphisms ф₁：E|ʏ → E'|ʏ，ф₂：F|ʏ → F'|ʏ such that σ'◦ ф₁＝ф₂◦σ . We say that A＝(E，F，σ)，A'＝(E'，F'，σ') are equivalent if ∃B，B' ∈ 𝓛 (X，Y) such that

A ⨁ B＝A' ⨁ B.

Denote the set of equivalence classes [E，F，σ] by L(X，Y)，and this is an abelian group under the obviously defined direct sum.

Lemma 6 There erists αn equiυαlence of functors χ：L(X，Y) → K(X，Y)， such that ωhen Y＝∅，[E，F，σ] ↦[E] – [F].

We only sketch a proof. Given an element [V₀，V₁，σ] ∈ L₁(X，Y) we associate to it an element χ([V₀，V₁，σ]) ∈ K(X，Y). Set Xₖ＝X × {k} for k＝0，1 and consider the space Z＝X₀ ∪ʏ X₁ obtained from the disjoint union X₀ ∪ X₁ by identifying y × {0} with y × {1} for all y ∈ Y. The natural sequence

ⱼ* ᵢ*

0 → K(Z，X₁) → K(Z) → K(X₁) → 0

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is split exact since there is an obvious retraction ρ：Z → X₁. Furthermore，there is

≈

an isomorphism φ：K(Z，X₁) → K(X，Y). From our element [V₀，V₁，σ] we define a vector bundle W over Z by setting W[xₖ ≣ Vₖ and identifying over Y via the isomorphism

a. Setting W₁ ≡ ρ*(V₁) we have [W]– [W₁] ∈ ker(i*). Hence，there is a unique ele-ment χ([V₀，V₁，σ]) ∈ K(X，Y) with j*φ⁻¹χ([V₀，V₁，σ])＝[W] – [W₁]. This defines the homomorphism χ：L(X，Y) → K(X，Y). ▢

In the discussion above，we are requiring that spaces are compact and CW. But this is not true even for (T*M，T*M₀)，where M is a compact manifold and T*M₀；＝ T*M–zero section. But if we fix a Riemannian metric on T*M and consider bundles D*M，S*M ⊂ T*M with fiber unit solid balls and unit spheres， (D*M，S*M) becomes a CW-pair homotopy equivalent to (T*M，T*M₀). Th(T*M)：＝ D*M/S*M is called the Thom spαce of that bundle.

1.7 Thom isomorphism in K-theory

Recall the Thom isomorphism theorem, stating that for an oriented real rank n bundle π：E → B，there exists a unique class u ∈ Hⁿ (E，E₀；ℤ) such that for all k，we have the Thom isomorphism ф：Hᵏ (B；ℤ) → Hⁿ⁺ᵏ (E，E₀；ℤ)，x ↦(π*x)∪u.

There is a similar version in K-theory.

Theorem 9 For α compler υector bundle π：E → M oυer α compαct spαce M，ωe hαυe αn isomorphism

ψ：K(M) → K(E，E₀)，α ↦ π*α · d(π*(∧*(E))).

Firstly，we explain the element d(π*(∧*(E))) ∈ K(E，E₀). Consider mappings between vector bundles over E，

фᵢ：π*(∧ⁱE) → π*(∧ⁱ⁺¹E)，(ω，υ) ∈ π*(∧ⁱE) ↦ (ω，ω∧υ)

where ω ∈ E. When restricted to E₀， these mappings form an exact sequence，as is easily verified. We claim that these mappings determine a unique element d(π*(∧*(E))) ∈ K(E，E₀).

Let's generalize the disgussion in the last section. Assume Y ⊂ X is a closed subspace. For each integer n ≥ 1，consider the set 𝓛 ₙ (X，Y) of elements V＝(V₀，V₁，. . .，Vₙ；σ₁，. . .，σₙ) where V₀，. . .，Vₙ are vector bundles on X and where

σ₁ σ₂ σₙ

0→V₀|ʏ → V₁|ʏ → · · · → Vₙ|ʏ → 0

is an exact sequence of bundle maps for the restriction of these bundles to Y. Two such elements V＝(V₀，. . .，Vₙ；σ₁，. . .，σₙ) and V'＝(V'₀，. . .，V'ₙ；σ'₁，. . .，σ'ₙ) are said to be isomorphic if there are bundle isomorphisms φᵢ：Vᵢ → V'ᵢ over X such that everything commutes.

An element V＝(V₀，. . .，Vₙ；σ₁，. . .，σₙ) is said to be elementary if there is an i such that Vᵢ＝Vᵢ₋₁，σᵢ＝id and Vⱼ＝{0} for j ≠ i or i – 1. There is an operation of direct sum ⨁ defined on the set 𝓛 ₙ (X，Y) in the obvious way. Two elements V，V' ∈ 𝓛 ₙ(X，Y) are defined to be equivalent if there exist elementary elements E₁，. . .，Eₖ，F₁，. . .，Fₗ ∈ 𝓛 ₙ (X，Y) and an isomorphism

V ⨁ E₁⨁ · · · ⨁ Eₖ ≅ V' ⨁ F₁ ⨁ · · · ⨁ Fₗ

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