Appendix: Paracompactness

P

roof: A preliminary observation is that X is normal. To show this, it suffices to find a map f : X

→

I with f (A)= 0 and f (B) = 1 for any two disjoint closed sets A and B . Such an f can be constructed inductively over the X_i’s, using normality of the X_i’s.

For the induction step one has f defined on the closed set X_i∪(A∩X_i+1)∪(B ∩X_i+1) and one extends over X_i+1 by Tietze’s theorem.

To prove that X is paracompact, let an open cover {U_α} be given. Since X_i is compact Hausdorff, there is a finite partition of unity {ϕ_ij} on X_i subordinate to {Uα∩ Xi}. Using normality of X , extend each ϕij to a map ϕ_ij: X

→

I with support in the same U_α. Let σ_i =P

jϕ_ij. This sum is 1 on X_i, so if we normalize each ϕ_ij by dividing it by max{¹/₂, σ_i}, we get new maps ϕij with σ_i = 1 in a neighborhood V_i of X_i. Let ψ_ij= max{0, ϕ_ij−P

k<iσ_k}. Since 0 ≤ ψ_ij ≤ ϕ_ij, the collection {ψ_ij} is subordinate to {Uα}. In Vi all ψ_kj’s with k > i are zero, so each point of X has a neighborhood in which only finitely many ψ_ij’s are nonzero. For each x∈ X there is a ψ_ij with ψ_ij(x) > 0 , since if ϕ_ij(x) > 0 and i is minimal with respect to this condition, then ψ_ij(x)= ϕ_ij(x) . Thus when we normalize the collection {ψ_ij} by dividing byP

i,jψ_ij we obtain a partition of unity on X subordinate to{U_α}. tu

P

roposition 1.20. Every CW complex is paracompact.

P

roof: Given an open cover {Uα} of a CW complex X , suppose inductively that we have a partition of unity {ϕβ} on Xⁿ subordinate to the cover {Uα∩ Xⁿ}. For a cell e_γⁿ⁺¹ with characteristic map Φ_γ: Dⁿ⁺¹

→

^{X , {ϕβΦγ}} is a partition of unity on Sⁿ= ∂Dⁿ⁺¹. Since Sⁿis compact, only finitely many of these compositions ϕ_βΦ_γcan be nonzero, for fixed γ . We extend these functions ϕ_βΦγ over Dⁿ⁺¹ by the formula ρ_ε(r )ϕ_βΦγ(x) using ‘spherical coordinates’ (r , x)∈ I×Sⁿ on Dⁿ⁺¹, where ρ_ε: I

→

^I

is 0 on [0, 1−ε] and 1 on [1−ε/2, 1]. If ε= εγis chosen small enough, these extended functions ρ_εϕ_βΦ_γ will be subordinate to the cover {Φ⁻¹_γ (U_α)}. Let {ψ_γj} be a finite partition of unity on Dⁿ⁺¹subordinate to{Φ⁻¹_γ (U_α)}. Then {ρ_εϕ_βΦ_γ, (1−ρ_ε)ψ_γj} is a partition of unity on Dⁿ⁺¹subordinate to{Φ⁻¹γ (U_α)}. This partition of unity extends the partition of unity {ϕβΦγ} on Sⁿand induces an extension of {ϕβ} to a partition of unity defined on Xⁿ∪eⁿ⁺¹γ and subordinate to{Uα}. Doing this for all (n+1) cells e_γⁿ⁺¹ gives a partition of unity on Xⁿ⁺¹. The local finiteness condition continues to hold since near a point in Xⁿ only the extensions of the ϕ_β’s in the original partition of unity on Xⁿ are nonzero, while in a cell eⁿ⁺¹_γ the only other functions that can be nonzero are the ones coming from ψ_γj’s. After we make such extensions for all n , we obtain a partition of unity defined on all of X and subordinate to {U_α}. tu

Here is a technical fact about paracompact spaces that is occasionally useful:

L

emma 1.21. Given an open cover {Uα} of the paracompact space X , there is a countable open cover {Vk} such that each Vk is a disjoint union of open sets each contained in some U_α, and there is a partition of unity {ϕ_k} with ϕ_k supported in V_k.

P

roof: Let{ϕβ} be a partition of unity subordinate to {Uα}. For each finite set S of functions ϕ_β let V_S be the subset of X where all the ϕ_β’s in S are strictly greater than all the ϕ_β’s not in S . Since only finitely many ϕ_β’s are nonzero near any x∈ X , V_S is defined by finitely many inequalities among ϕ_β’s near x , so V_S is open. Also, V_S is contained in some U_α, namely, any U_α containing the support of any ϕ_β∈ S , since ϕ_β∈ S implies ϕβ> 0 on V_S. Let V_k be the union of all the open sets V_S such that S has k elements. This is clearly a disjoint union. The collection {V_k} is a cover of X since if x∈ X then x ∈ V_S where S= { ϕ_β| ϕ_β(x) > 0}.

For the second statement, let {ϕγ} be a partition of unity subordinate to the cover {Vk}, and let ϕk be the sum of those ϕ_γ’s supported in V_k but not in V_j for

j < k . tu

Exercises

1. Show that the projection V_n(R^k)

→

^Gn(Rk) is a fiber bundle with fiber O(n) by showing that it is the orthonormal n frame bundle associated to the vector bundle E_n(R^k)

→

^{Gn(Rk) .}

bundles over a fixed base space X into the addition operation in a group. There are two slightly different ways of doing this, producing, in the case of complex vector bundles, groups K(X) and eK(X) with K(X)≈ eK(X)⊕Z, and for real vector bundles, groups KO(X) and gKO(X) with KO(X)≈ gKO(X)⊕Z. Complex K–theory turns out to be somewhat simpler than real K–theory, so we concentrate on this case in the present chapter.

Computing eK(X) even for simple spaces X requires some work. The case X= Sⁿ is the Bott Periodicity Theorem, which gives isomorphisms eK(Sⁿ)≈ eK(Sⁿ⁺²) for all n , and more generally eK(X) ≈ eK(S²X) where S²X is the double suspension of X . This is a deep theorem, so it is not surprising that it has applications of real substance.

We give some of these in§2.3, notably:

(1) The nonexistence of division algebras over R in dimensions other than 1, 2, 4, and 8 , the dimensions of the real and complex numbers, quaternions, and Cayley octonions.

(2) The nonparallelizability of spheres other than S¹, S³, and S⁷.

The proof of the Bott Periodicity Theorem divides into two parts. The first is the hard technical work, proving an isomorphism K(X×S²)≈ K(X)⊗K(S²) . This takes about ten pages, forming the bulk of§2.1. The other half is easier, being more formal in nature, and this is contained in§2.2 where the cohomological aspects of K(X) are the main focus.

2.1. The Functor K(X)

Since we shall be dealing exclusively with complex vector bundles in this chapter, let us take ‘vector bundle’ to mean ‘complex vector bundle’ unless otherwise specified.

Base spaces will always be assumed to be compact Hausdorff, so that the results of the preceding chapter which have this hypothesis will be available to us.

For the purposes of K–theory it is convenient to take a slightly broader defini-tion of ‘vector bundle’ which allows the fibers of a vector bundle p : E

→

X to be vec-tor spaces of different dimensions. We still assume local trivializations of the form h : p⁻¹(U)

→

^U×Cn, so the dimensions of fibers must be locally constant over X , but if X is disconnected the dimensions of fibers need not be globally constant.

Consider vector bundles over a fixed base space X . The trivial n dimensional vector bundle we write as εⁿ

→

X . Define two vector bundles E₁ and E₂ over X to be stably isomorphic, written E₁≈sE₂, if E₁⊕εⁿ≈ E2⊕εⁿ for some n . In a similar vein we set E₁∼ E₂ if E₁⊕^εm≈ E₂⊕^εn for some m and n . It is easy to see that both ≈_s and ∼ are equivalence relations. On equivalence classes of either sort the operation of direct sum is well-defined, commutative, and associative. A zero element is the class of ε⁰.

P

roposition 2.1. If X is compact Hausdorff, then the set of∼ equivalence classes of vector bundles over X forms an abelian group with respect to ⊕.

This group is called eK(X) .

P

roof: Only the existence of inverses needs to be shown, which we do by showing that for each vector bundle π : E

→

X there is a bundle E⁰

→

X such that E⊕^E0≈ ε^m for some m . If all the fibers of E have the same dimension, this is Proposition 1.4.

In the general case let X_i = { x ∈ X || dim π⁻¹(x)= i }. These Xi’s are disjoint open sets in X , hence are finite in number by compactness. By adding to E a bundle which over each X_i is a trivial bundle of suitable dimension we can produce a bundle whose

fibers all have the same dimension. tu

For the direct sum operation on≈s equivalence classes, only the zero element, the class of ε⁰, can have an inverse since E⊕E⁰≈sε⁰ implies E⊕E⁰⊕εⁿ≈ εⁿ for some n , which can only happen if E and E⁰ are 0 dimensional. However, even though inverses do not exist, we do have the cancellation property that E₁⊕^E2 ≈_s E₁⊕^E3

implies E₂ ≈_s E₃ over a compact base space X , since we can add to both sides of E₁⊕E₂≈sE₁⊕E₃ a bundle E⁰₁ such that E₁⊕E⁰₁≈ εⁿ for some n .

Just as the positive rational numbers are constructed from the positive integers by forming quotients a/b with the equivalence relation a/b= c/d iff ad = bc , so we can form for compact X an abelian group K(X) consisting of formal differences E−E⁰ of vector bundles E and E⁰ over X , with the equivalence relation E₁− E1⁰ = E2− E⁰2

iff E₁⊕E⁰₂≈s E₂⊕E₁⁰. Verifying transitivity of this relation involves the cancellation

property, which is why compactness of X is needed. With the obvious addition rule (E₁−E⁰₁)+(E₂−E⁰₂)= (E₁⊕^E2)−(E₁⁰⊕^E02) , K(X) is then a group. The zero element is the equivalence class of E− E for any E , and the inverse of E − E⁰ is E⁰− E . Note that every element of K(X) can be represented as a difference E−εⁿ since if we start with E− E⁰ we can add to both E and E⁰ a bundle E⁰⁰ such that E⁰⊕E⁰⁰≈ εⁿ for some n . There is a natural homomorphism K(X)

→

K(X) sending E^e − εⁿ to the ∼ class of E . This is well-defined since if E− εⁿ= E⁰− ε^m in K(X) , then E⊕ε^m≈sE⁰⊕εⁿ, hence E∼ E⁰. The map K(X)

→

K(X) is obviously surjective, and its kernel consists of^e elements E−εⁿwith E∼ ε⁰, hence E≈_sε^m for some m , so the kernel consists of the elements of the form ε^m− εⁿ. This subgroup {ε^m− εⁿ} of K(X) is isomorphic to Z.

In fact, restriction of vector bundles to a basepoint x₀∈ X defines a homomorphism K(X)

→

^K(x0) ≈ Z which restricts to an isomorphism on the subgroup {ε^m− εⁿ}.

Thus we have a splitting K(X)≈ eK(X)⊕Z, depending on the choice of x₀. The group K(X) is sometimes called reduced, to distinguish it from K(X) .e

在文檔中 1.2. Classifying Vector Bundles (頁 39-44)