The Splitting Principle - 1.2. Classifying Vector Bundles

Now we can use the operations ψ² and ψ³ and the relation ψ²ψ³= ψ⁶= ψ³ψ² to prove Adams’ theorem.

P

roof of Theorem 2.19: The definition of the Hopf invariant of a map f : S⁴ⁿ⁻¹

→

^S²ⁿ

involved elements α, β∈ eK(C_f) . By Proposition 2.21, ψ^k(α)= k²ⁿα since α is the image of a generator of eK(S⁴ⁿ) . Similarly, ψ^k(β) = kⁿβ+ µkα for some µ_k ∈ Z.

Therefore

ψ^kψ^`(β)= ψ^k(`ⁿβ+ µ_`α)= kⁿ`ⁿβ+ (k²ⁿµ_`+ `ⁿµ_k)α

Since ψ^kψ^`= ψ^k`= ψ^`ψ^k, the coefficient k²ⁿµ_`+ `ⁿµ_k of α is unchanged when k and ` are switched. This gives the relation

k²ⁿµ_`+ `ⁿµ_k= `²ⁿµ_k+ kⁿµ_`, or (k²ⁿ− kⁿ)µ_`= (`²ⁿ− `ⁿ)µ_k

By property (6) of ψ², we have ψ²(β)≡ β²mod 2 . Since β² = hα with h the Hopf invariant of f , the formula ψ²(β)= 2ⁿβ+ µ₂α implies that µ₂≡ h mod 2, so µ₂ is odd if we assume h= ±1. By the preceding displayed formula we have (2²ⁿ−2ⁿ)µ₃= (3²ⁿ− 3ⁿ)µ₂, or 2ⁿ(2ⁿ− 1)µ3= 3ⁿ(3ⁿ− 1)µ2, so 2ⁿ divides 3ⁿ(3ⁿ− 1)µ2. Since 3ⁿ and µ₂ are odd, 2ⁿ must then divide 3ⁿ− 1. The proof is completed by the following

elementary number theory fact. tu

L

emma 2.22. If 2ⁿ divides 3ⁿ− 1 then n = 1, 2, or 4.

P

roof: Write n= 2^`m with m odd. We will show that the highest power of 2 dividing 3ⁿ− 1 is 2 for ` = 0 and 2^`+2 for ` > 0 . This implies the lemma since if 2ⁿ divides 3ⁿ− 1, then by this fact, n ≤ ` + 2, hence 2^`≤ 2^`m= n ≤ ` + 2, which implies ` ≤ 2 and n≤ 4. The cases n = 1, 2, 3, 4 can then be checked individually.

We find the highest power of 2 dividing 3ⁿ− 1 by induction on `. For ` = 0 we have 3ⁿ− 1 = 3^m− 1 ≡ 2 mod 4 since 3 ≡ −1 mod 4 and m is odd. In the next case ` = 1 we have 3ⁿ− 1 = 3^2m− 1 = (3^m− 1)(3^m+ 1). The highest power of 2 dividing the first factor is 2 as we just showed, and the highest power of 2 dividing the second factor is 4 since 3^m+ 1 ≡ 4 mod 8 because 3² ≡ 1 mod 8 and m is odd. So the highest power of 2 dividing the product (3^m− 1)(3^m+ 1) is 8. For the inductive step of passing from ` to `+ 1 with ` ≥ 1, or in other words from n to 2n with n even, write 3²ⁿ− 1 = (3ⁿ− 1)(3ⁿ+ 1). Then 3ⁿ+ 1 ≡ 2 mod 4 since n is even, so the highest power of 2 dividing 3ⁿ+ 1 is 2. Thus the highest power of 2 dividing 3²ⁿ− 1 is twice the highest power of 2 dividing 3ⁿ− 1. tu

P

roposition 2.23. If X is a finite cell complex with n cells, then K^∗(X) is a finitely generated group with at most n generators. If all the cells of X have even dimension then K¹(X)= 0 and K⁰(X) is free abelian with one basis element for each cell.

The phrase ‘finite cell complex’ would normally mean ‘finite CW complex’ but we can take it to be something slightly more general: a space built from a finite discrete set by attaching a finite number of cells in succession, with no conditions on the dimen-sions of these cells, so cells are not required to attach only to cells of lower dimension.

Finite cell complexes are always homotopy equivalent to finite CW complexes (by de-forming each successive attaching map to be cellular) so the only advantages of finite cell complexes are technical. In particular, it is easy to see that a space is a finite cell complex if it is a fiber bundle over a finite cell complex with fibers that are also finite cell complexes. This is shown in Proposition 2.28 in a brief appendix to this section.

It implies that the splitting principle can be applied staying within the realm of finite cell complexes.

P

roof: We show this by induction on the number of cells. The complex X is obtained from a subcomplex A by attaching a k cell, for some k . For the pair (X, A) we have an exact sequence eK^∗(X/A)

-→

^K^e^∗^(X)

-→

^K^e^∗(A) . Since X/A = S^k, we have Ke^∗(X/A)≈ Z, and exactness implies that eK^∗(X) requires at most one more generator than eK^∗(A) .

The first term of the exact sequence K¹(X/A)

→

^K¹^(X)

→

^K¹(A) is zero if all cells of X are of even dimension, so induction on the number of cells implies that K¹(X)= 0. Then there is a short exact sequence 0

→

^K^e⁰^(X/A)

→

^K^e⁰^(X)

→

^K^e⁰^(A)

→

⁰

with eK⁰(X/A)≈ Z. By induction eK(A) is free, so this sequence splits, hence K⁰(X)≈ Z⊕K⁰(A) and the final statement of the proposition follows. tu This proposition applies in particular toCPⁿ, which has a cell structure with one cell in each dimension 0, 2, 4,··· , 2n, so K¹(CPⁿ)= 0 and K⁰(CPⁿ)≈ Zⁿ⁺¹. The ring structure is as simple as one could hope for:

P

roposition 2.24. K(CPⁿ) is the quotient ringZ[L]/(L−1)ⁿ⁺¹ where L is the canon-ical line bundle overCPⁿ.

Thus by the change of variable x= L−1 we see that K(CPⁿ) is the truncated poly-nomial ringZ[x]/(xⁿ⁺¹) , with additive basis 1, x,··· , xⁿ. It follows that 1, L,··· , Lⁿ is also an additive basis.

P

roof: The exact sequence for the pair (CPⁿ,CPⁿ⁻¹) gives a short exact sequence 0

-→

^K(CPⁿ^,^CPⁿ⁻¹⁾

-→

^K(CPⁿ⁾

---→

^ρ ^K(CPⁿ⁻¹⁾

-→

⁰

We shall prove:

(a_n) (L− 1)ⁿ generates the kernel of the restriction map ρ .

Hence if we assume inductively that K(CPⁿ⁻¹)= Z[L]/(L − 1)ⁿ, then (a_n) and the preceding exact sequence imply that {1, L − 1, ··· , (L − 1)ⁿ} is an additive basis for K(CPⁿ) . Since (L− 1)ⁿ⁺¹ = 0 in K(CPⁿ) by (a_n+1) , it follows that K(CPⁿ) is the quotient ringZ[L]/(L − 1)ⁿ⁺¹, completing the induction.

A reason one might expect (a_n) to be true is that the kernel of ρ can be identi-fied with K(CPⁿ,CPⁿ⁻¹)= eK(S²ⁿ) by the short exact sequence, and Bott periodicity implies that the n fold reduced external product of the generator L− 1 of eK(S²) with itself generates eK(S²ⁿ) . To make this rough argument into a proof we will have to relate the external product eK(S²)⊗ ··· ⊗ eK(S²)

→

^K(S^e ²ⁿ) to the ‘internal’ product K(CPⁿ)⊗ ··· ⊗K(CPⁿ)

→

^K(CPⁿ^{) .}

The space CPⁿ is the quotient of the unit sphere S²ⁿ⁺¹ in Cⁿ⁺¹ under multipli-cation by scalars in S¹⊂ C. Instead of S²ⁿ⁺¹ we could equally well take the boundary of the product D²₀× ··· ×D²_nwhere D²_i is the unit disk in the i^thcoordinate of Cⁿ⁺¹, and we start the count with i= 0 for convenience. Then we have

∂(D²₀× ··· ×D²_n)=S

i(D²₀× ··· ×∂D_i²× ··· ×D²_n)

The action of S¹ by scalar multiplication respects this decomposition. The orbit space of D²₀× ··· ×∂D²i× ··· ×D²n under the action is a subspace C_i⊂ CPⁿ homeomorphic to the product D₀²× ··· ×Dn² with the factor D_i² deleted. Thus we have a decomposi-tion CPⁿ=S

iC_i with each C_i homeomorphic to D²ⁿ and with C_i∩ C_j = ∂C_i∩ ∂C_j for i≠ j .

Consider now C₀ = D²1× ··· ×D²n. Its boundary is decomposed into the pieces

∂_iC₀ = D²1× ··· ×∂D²i× ··· ×D²n. The inclusions (D_i², ∂D²_i)⊂ (C0, ∂_iC₀)⊂ (CPⁿ, C_i) give rise to a commutative diagram

K( 1², ∂D1²) K(Dn², ∂Dn²)

−−−−−−−−−−→

−−−−−→ −−−−−→

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→

−−−−−−−−−−−−−−−−−→

− − − −−−−−

−−→ −−→− −→ −−→

^≈

−−→ − →

_≈

...

K( ₀, ∂₁C0)

...

K(C₀,∂nC₀) K(C0,∂C0)

K( P ,C1) n

... ...

≈ ≈

≈

C K( P C ) K( P ,C1 n

C n C ) )

n, C

K(_CPⁿ)

...

_K₍_C_Pⁿ₎ _K₍_C_Pⁿ₎

∪ ∪ K(CPⁿ,CP^{n 1}

-⊗ ⊗

⊗ ⊗

where the maps from the first column to the second are the n fold products. The upper map in the middle column is an isomorphism because the inclusion C₀

>

^CPⁿ

induces a homeomorphism C₀/∂C₀≈ CPⁿ/(C₁∪···∪Cn) . TheCPⁿ⁻¹at the right side of the diagram sits in CPⁿ in the last n coordinates ofCⁿ⁺¹, so is disjoint from C₀, hence the quotient mapCPⁿ/CPⁿ⁻¹

→

^CPⁿ^/(C¹^{∪···∪C}ⁿ) is a homotopy equivalence.

The element x_i∈ K(CPⁿ, C_i) mapping downward to L−1 ∈ K(CPⁿ) maps upward to a generator of K(C₀, ∂_iC₀) ≈ K(Di², ∂D²_i) . By commutativity of the diagram, the product x₁··· xn then generates K(CPⁿ, C₁∪ ··· ∪ Cn) . This means that (L− 1)ⁿ

generates the image of the map K(CPⁿ,CPⁿ⁻¹)

→

^K(CPⁿ) , which equals the kernel of

ρ , proving (a_n) . tu

SinceCPⁿis the union of the n+1 balls Ci, Example 2.13 shows that all products of n+ 1 elements of eK(CPⁿ) must be zero, in particular (L− 1)ⁿ⁺¹= 0. But as we have just seen, (L− 1)ⁿ is nonzero, so the result in Example 2.13 is best possible in terms of the degree of nilpotency.

Now we come to the Leray-Hirsch theorem for K–theory, which will be the major theoretical ingredient in the proof of the splitting principle:

T

heorem 2.25. Let p : E

→

B be a fiber bundle with E and B compact Hausdorff and with fiber F such that K^∗(F ) is free. Suppose that there exist classes c₁,··· , ck ∈ K^∗(E) that restrict to a basis for K^∗(F ) in each fiber F . If either

(a) B is a finite cell complex, or

(b) F is a finite cell complex having all cells of even dimension, then K^∗(E) , as a module over K^∗(B) , is free with basis{c1,··· , ck}.

Here the K^∗(B) module structure on K^∗(E) is defined by β· γ = p^∗(β)γ for β ∈ K^∗(B) and γ ∈ K^∗(E) . Another way to state the conclusion of the theorem is to say that the mapΦ : K^∗(B)⊗K^∗(F )

→

^K^∗^{(E) ,} ^Φ(^Pⁱ^bⁱ^⊗ⁱ^∗^(cⁱ⁾⁾⁼^Pⁱ^p^∗^(bⁱ^)cⁱ ^{for i}

the inclusion F

>

E , is an isomorphism.

In the case of the product bundle E= F ×B the classes c_i can be chosen to be the pullbacks under the projection E

→

F of a basis for K^∗(F ) . The theorem then asserts that the external product K^∗(F )⊗K^∗(B)

→

^K^∗^(F×B) is an isomorphism.

For most of our applications of the theorem either case (a) or case (b) will suffice.

The proof of (a) is somewhat simpler than (b), and we include (b) mainly to obtain the splitting principle for vector bundles over arbitrary compact Hausdorff base spaces.

P

roof: For a subspace B⁰⊂ B let E⁰= p⁻¹(B⁰) . Then we have a diagram

−−→ −−→

−−→ −−−−−→

−−−−−→

^∗ ⁰

∗

⊗

Φ Φ Φ

B B K ( ) ( )

, K ( )^∗ F ^{K ( )}^∗ ^B ^⊗^{K ( )}^∗ ^F

−−−−−→

^{K (}^∗ ^B⁰⁾^⊗^{K ( )}^∗ ^F

− − − − − →

−−−−−−−−−−−−−−−→

−−−−−−−−−−−−→

^{K (}^∗ ^E^,^E⁰⁾ ^{K ( )}^∗ ^E

−−−−−−−−−−−−−−−−−→

^{K (}^∗ ^E⁰⁾

− − − −−−−−−− − →

where the left-handΦ is defined by the same formula Φ(P

ib_i⊗i^∗(c_i))=P

ip^∗(b_i)c_i, but with p^∗(b_i)c_ireferring now to the relative product K^∗(E, E⁰)×K^∗(E)

→

^K^∗^{(E, E}⁰^{) .}

The right-hand Φ is defined using the restrictions of the ci’s to the subspace E⁰. To see that the diagram (∗) commutes, we can interpolate between its two rows the row

-→

^K^∗^{(E, E}⁰⁾^⊗K^∗^{(F )}

-→

^K^∗^(E)^⊗K^∗^{(F )}

-→

^K^∗^(E⁰⁾^⊗K^∗^{(F )}

-→

by factoring Φ as the compositionP

ib_i⊗i^∗(c_i)

,

^Pⁱ^p^∗^(bⁱ⁾^⊗ⁱ^∗^(cⁱ⁾

,

^Pⁱ^p^∗^(bⁱ^)cⁱ^.

The upper squares of the enlarged diagram then commute trivially, and the lower squares commute by Proposition 2.15. The lower row of the diagram is of course

exact. The upper row is also exact since we assume K^∗(F ) is free, and tensoring an exact sequence with a free abelian group preserves exactness, the result of the tensoring operation being simply to replace the given exact sequence by the direct sum of a number of copies of itself.

The proof in case (a) will be by a double induction, first on the dimension of B , then within a given dimension, on the number of cells. The induction starts with the trivial case that B is zero-dimensional, hence a finite discrete set. For the induction step, suppose B is obtained from a subcomplex B⁰ by attaching a cell eⁿ, and let E⁰= p⁻¹(B⁰) as above. By induction on the number of cells of B we may assume the right-hand Φ in (∗) is an isomorphism. If the left-hand Φ is also an isomorphism, then the five-lemma will imply that the middle Φ is an isomorphism, finishing the induction step.

Let ϕ : (Dⁿ, Sⁿ⁻¹)

→

^{(B, B}⁰) be a characteristic map for the attached n cell. Since Dⁿ is contractible, the pullback bundle ϕ^∗(E) is a product, and so we have a com-mutative diagram

−−→

−−→ −−−−−→

∗ ⊗

Φ Φ Φ

B B

K ( , ) K ( )^∗ F K (^∗ D S, )⊗K ( )^∗ F

−−−−−−−−−−−−→

0 0

∗ E E

K ( , ) K^∗

(

^∗(E),

−−−−−−−−−−−−−→

K (∗

≈

≈ ϕ ϕ^∗(E )

)

≈ × ×

n n 1

-, F

Dⁿ S^{n 1}^- )

The two horizontal maps are isomorphisms since ϕ restricts to a homeomorphism on the interior of Dⁿ, hence induces homeomorphisms B/B⁰≈ Dⁿ/Sⁿ⁻¹ and E/E⁰≈ ϕ^∗(E)/ϕ^∗(E⁰) . Thus the diagram reduces the proof to showing that the right-hand Φ, involving the product bundle Dⁿ×F

→

^Dⁿ, is an isomorphism.

Consider the diagram (∗) with (B, B⁰) replaced by (Dⁿ, Sⁿ⁻¹) . We may assume the right-hand Φ in (∗) is an isomorphism since Sⁿ⁻¹ has smaller dimension than the original cell complex B . The middle Φ is an isomorphism by the case of zero-dimensional B since Dⁿdeformation retracts to a point. Therefore by the five-lemma the left-hand Φ in (∗) is an isomorphism for (B, B⁰)= (Dⁿ, Sⁿ⁻¹) . This finishes the proof in case (a).

In case (b) let us first prove the result for a product bundle E= F ×B . In this case Ψ is just the external product, so we are free to interchange the roles of F and B . Thus we may use the diagram (∗) with F an arbitrary compact Hausdorff space and B a finite cell complex having all cells of even dimension, obtained by attaching a cell eⁿ to a subcomplex B⁰. The upper row of (∗) is then an exact sequence since it is obtained from the split short exact sequence 0

→

^K^∗^{(B, B}⁰⁾

→

^K^∗^(B)

→

^K^∗^(B⁰⁾

→

^{0 by}

tensoring with the fixed group K^∗(F ) . If we can show that the left-hand Φ in (∗) is an isomorphism, then by induction on the number of cells of B we may assume the right-hand Φ is an isomorphism, so the five-lemma will imply that the middle Φ is also an isomorphism.

To show the left-handΦ is an isomorphism, note first that B/B⁰= Sⁿ so we may

as well take the pair (B, B⁰) to be (Dⁿ, Sⁿ⁻¹) . Then the middle Φ in (∗) is obviously an isomorphism, so the left-hand Φ will be an isomorphism iff the right-hand Φ is an isomorphism. When the sphere Sⁿ⁻¹ is even-dimensional we have already shown that Φ is an isomorphism in the remarks following the proof of Lemma 2.17, and the same argument applies also when the sphere is odd-dimensional, since K¹ of an odd-dimensional sphere is K⁰ of an even-dimensional sphere.

Now we turn to case (b) for nonproducts. The proof will once again be inductive, but this time we need a more subtle inductive statement since B is just a compact Hausdorff space, not a cell complex. Consider the following condition on a compact subspace U⊂ B :

For all compact V ⊂ U the map Φ : K^∗(V )⊗K^∗(F )

→

^K^∗^(p⁻¹(V )) is an isomor-phism.

If this is satisfied, let us call U good. By the special case already proved, each point of B has a compact neighborhood U that is good. Since B is compact, a finite number of these neighborhoods cover B , so by induction it will be enough to show that if U₁ and U₂ are good, then so is U₁∪ U₂.

A compact V ⊂ U₁∪ U₂ is the union of V₁= V ∩ U₁ and V₂= V ∩ U₂. Consider the diagram like (∗) for the pair (V , V2) . Since K^∗(F ) is free, the upper row of this diagram is exact. Assuming U₂ is good, the map Φ is an isomorphism for V2, so Φ will be an isomorphism for V if it is an isomorphism for (V , V₂) . The quotient V /V₂ is homeomorphic to V₁/(V₁∩ V₂) so Φ will be an isomorphism for (V, V₂) if it is an isomorphism for (V₁, V₁∩ V₂) . Now look at the diagram like (∗) for (V₁, V₁∩ V₂) . Assuming U₁ is good, the mapsΦ are isomorphisms for V1 and V₁∩ V2. Hence Φ is an isomorphism for (V₁, V₁∩ V2) , and the induction step is finished. tu

E

xample 2.26. Let E

→

X be a vector bundle with fibers Cⁿ and compact base X . Then we have an associated projective bundle p : P (E)

→

X with fibersCPⁿ⁻¹, where P (E) is the space of lines in E , that is, one-dimensional linear subspaces of fibers of E . Over P (E) there is the canonical line bundle L

→

P (E) consisting of the vectors in the lines of P (E) . In each fiber CPⁿ⁻¹ of P (E) the classes 1, L,··· , Lⁿ⁻¹ in K^∗(P (E)) restrict to a basis for K^∗(CPⁿ⁻¹) by Proposition 2.24. From the Leray-Hirsch theorem we deduce that K^∗(P (E)) is a free K^∗(X) module with basis 1, L,··· , Lⁿ⁻¹.

P

roof of the Splitting Principle: In the preceding example, the fact that 1 is among the basis elements implies that p^∗: K^∗(X)

→

^K^∗(P (E)) is injective. The pullback bundle p^∗(E)

→

P (E) contains the line bundle L as a subbundle, hence splits as L⊕E⁰ for E⁰

→

P (E) the subbundle of p^∗(E) orthogonal to L with respect to some choice of inner product. Now repeat the process by forming P (E⁰) , splitting off another line bundle from the pullback of E⁰ over P (E⁰) . Note that P (E⁰) is the space of pairs of orthogonal lines in fibers of E . After a finite number of repetitions we obtain the flag bundle F (E)

→

X described at the end of§1.1, whose points are n tuples of

orthogonal lines in fibers of E , where n is the dimension of E . (If the fibers of E have different dimensions over different components of X , we do the construction for each component separately.) The pullback of E over F (E) splits as a sum of line bundles, and the map F (E)

→

X induces an injection on K^∗ since it is a composition

of maps with this property. tu

In the preceding Example 2.26 we saw that K^∗(P (E)) is free as a K^∗(X) module, with basis 1, L,··· , Lⁿ⁻¹. In order to describe the multiplication in K^∗(P (E)) one therefore needs only a relation expressing Lⁿ in terms of lower powers of L . Such a relation can be found as follows. The pullback of E over P (E) splits as L⊕E⁰ for some bundle E⁰ of dimension n− 1, and the desired relation will be λⁿ(E⁰)= 0. To compute λⁿ(E⁰)= 0 we use the formula λ_t(E)= λ_t(L)λ_t(E⁰) in K^∗(P (E))[t] , where to simplify notation we let ‘ E ’ also denote the pullback of E over P (E) . The equation λ_t(E) = λt(L)λ_t(E⁰) can be rewritten as λ_t(E⁰) = λt(E)λ_t(L)⁻¹ where λ_t(L)⁻¹ = P

i(−1)ⁱLⁱtⁱ since λ_t(L) = 1 + Lt . Equating coefficients of tⁿ in the two sides of λ_t(E⁰)= λ_t(E)λ_t(L)⁻¹, we get λⁿ(E⁰)=P

i(−1)ⁿ⁻ⁱλⁱ(E)Lⁿ⁻ⁱ. The relation λⁿ(E⁰)= 0 can be written as P

i(−1)ⁱλⁱ(E)Lⁿ⁻ⁱ = 0, with the coefficient of Lⁿ equal to 1 , as desired. The result can be stated in the following form:

P

roposition 2.27. For an n dimensional vector bundle E

→

X the ring K(P (E)) is isomorphic to the quotient ring K^∗(X)[L]/ P

i(−1)ⁱλⁱ(E)Lⁿ⁻ⁱ

. tu

For example when X is a point we have P (E)= CPⁿ⁻¹and λⁱ(E)= C^k for k=

n i

, so the polynomialP

i(−1)ⁱλⁱ(E)Lⁿ⁻ⁱbecomes (L−1)ⁿand we see that the proposition generalizes the isomorphism K^∗(CPⁿ⁻¹)≈ Z[L]/(L − 1)ⁿ) .

Appendix: Finite Cell Complexes

As we mentioned in the remarks following Proposition 2.23 it is convenient for purposes of the splitting principle to work with spaces slightly more general than finite CW complexes. By a finite cell complex we mean a space which has a finite filtration X₀ ⊂ X1⊂ ··· ⊂ Xk = X where X0 is a finite discrete set and X_i+1 is obtained from X_i by attaching a cell eⁿⁱ via a map ϕ_i: Sⁿⁱ⁻¹

→

^Xⁱ^{. Thus X}i+1 is the quotient space of the disjoint union of X_i and a disk Dⁿⁱ under the identifications x ∼ ϕ_i(x) for x∈ ∂Dⁿⁱ = Sⁿⁱ⁻¹.

P

roposition 2.28. If p : E

→

B is a fiber bundle whose fiber F and base B are both finite cell complexes, then E is also a finite cell complex, whose cells are products of cells in B with cells in F .

P

roof: Suppose B is obtained from a subcomplex B⁰ by attaching a cell eⁿ. By induc-tion on the number of cells of B we may assume that p⁻¹(B⁰) is a finite cell complex.

If Φ : Dⁿ

→

B is a characteristic map for eⁿ then the pullback bundle Φ^∗(E)

→

^Dⁿ ^is

a product since Dⁿ is contractible. Since F is a finite cell complex, this means that

we may obtainΦ^∗(E) from its restriction over Sⁿ⁻¹ by attaching cells. Hence we may

obtain E from p⁻¹(B⁰) by attaching cells. tu

bundles E

→

B by some general rule which applies to all base spaces B . The four classical types of characteristic classes are:

1. Stiefel-Whitney classes w_i(E)∈ Hⁱ(B;Z2) for a real vector bundle E . 2. Chern classes c_i(E)∈ H²ⁱ(B;Z) for a complex vector bundle E . 3. Pontryagin classes p_i(E)∈ H⁴ⁱ(B;Z) for a real vector bundle E .

4. The Euler class e(E)∈ Hⁿ(B;Z) when E is an oriented n dimensional real vector bundle.

The Stiefel-Whitney and Chern classes are formally quite similar. Pontryagin classes can be regarded as a refinement of Stiefel-Whitney classes when one takes Z rather thanZ2 coefficients, and the Euler class is a further refinement in the orientable case.

Stiefel-Whitney and Chern classes lend themselves well to axiomatization since in most applications it is the formal properties encoded in the axioms which one uses rather than any particular construction of these classes. The construction we give, using the Leray-Hirsch theorem (proved in§4.D of [AT]), has the virtues of simplicity and elegance, though perhaps at the expense of geometric intuition into what prop-erties of vector bundles these characteristic classes are measuring. There is another definition via obstruction theory which does provide some geometric insights, and this will be described in the Appendix to this chapter.

3.1. Stiefel-Whitney and Chern Classes

Stiefel-Whitney classes are defined for real vector bundles, Chern classes for com-plex vector bundles. The two cases are quite similar, but for concreteness we shall em-phasize the real case, with occasional comments on the minor modifications needed to treat the complex case.

A technical point before we begin: We shall assume without further mention that all base spaces of vector bundles are paracompact, so that the fundamental results of Chapter 1 apply. For the study of characteristic classes this is not an essential restriction since one can always pass to pullbacks over a CW approximation to a given base space, and CW complexes are paracompact.

在文檔中 1.2. Classifying Vector Bundles (頁 69-78)