Axioms and Construction

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.

As in the real case, the formula in (b) for the total Chern classes can be rewritten in the form c_n(E₁⊕^E2)=P

i+j=nc_i(E₁)`^cj(E₂) , where c₀= 1.

P

roof of 3.1 and 3.2: Associated to a vector bundle π : E

→

B with fiber Rⁿ is the projective bundle P (π ) : P (E)

→

B , where P (E) is the space of all lines through the origin in all the fibers of E , and P (π ) is the natural projection sending each line in π⁻¹(b) to b∈ B . We topologize P(E) as a quotient of the complement of the zero section of E , the quotient obtained by factoring out scalar multiplication in each fiber.

Over a neighborhood U in B where E is a product U×Rⁿ, this quotient is U×RPⁿ⁻¹, so P (E) is a fiber bundle over B with fiberRPⁿ⁻¹.

We would like to apply the Leray-Hirsch theorem for cohomology with Z2 co-efficients to this bundle P (E)

→

B . To do this we need classes x_i ∈ Hⁱ(P (E);Z2) restricting to generators of Hⁱ(RPⁿ⁻¹;Z₂) in each fiber RPⁿ⁻¹ for i = 0, ··· , n − 1.

Recall from the proof of Theorem 1.8 that there is a map g : E

→

^R∞ that is a linear injection on each fiber. Projectivizing the map g by deleting zero vectors and then factoring out scalar multiplication produces a map P (g) : P (E)

→

^RP∞. Let α be a gen-erator of H¹(RP^∞;Z2) and let x= P(g)^∗(α)∈ H¹(P (E);Z2) . Then the powers xⁱ for i= 0, ··· , n − 1 are the desired classes x_i since a linear injection Rⁿ

→

^{R∞ induces}

an embeddingRPⁿ⁻¹

>

^RP∞ for which α pulls back to a generator of H¹(RPⁿ⁻¹;Z₂) , hence αⁱ pulls back to a generator of Hⁱ(RPⁿ⁻¹;Z2) . Note that any two linear injec-tions Rⁿ

→

^R∞ are homotopic through linear injections, so the induced embeddings RPⁿ⁻¹

>

^RP∞ of different fibers of P (E) are all homotopic. We showed in the proof of Theorem 1.8 that any two choices of g are homotopic through maps that are linear injections on fibers, so the classes xⁱ are independent of the choice of g .

The Leray-Hirsch theorem then says that H^∗(P (E);Z₂) is a free H^∗(B;Z₂) module with basis 1, x,··· , xⁿ⁻¹. Consequently, xⁿ can be expressed uniquely as a linear combination of these basis elements with coefficients in H^∗(B;Z2) . Thus there is a unique relation of the form

xⁿ+ w₁(E)xⁿ⁻¹+ ··· + w_n(E)· 1 = 0

for certain classes w_i(E)∈ Hⁱ(B;Z2) . Here w_i(E)xⁱ means P (π )^∗(w_i(E))`xⁱ, by the definition of the H^∗(B;Z2) module structure on H^∗(P (E);Z2) . For completeness we define w_i(E)= 0 for i > n and w₀(E)= 1.

To prove property (a), consider a pullback f^∗(E) = E⁰, fitting

into the diagram at the right. If g : E

→

^R∞ is a linear injection on

−−→

⁰

−−−−−→ −−→

0

E E

−−−−−→

B0 ^f B

f∼ π fibers then so is g ef , and it follows that P ( ef )^∗ takes the canonical π

class x= x(E) for P(E) to the canonical class x(E⁰) for P (E⁰) . Then P ( ef )^∗X

i

P (π )^∗ w_i(E)

`x(E)ⁿ⁻ⁱ

=X

i

P ( ef )^∗P (π )^∗ w_i(E)

`P ( ef )^∗ x(E)ⁿ⁻ⁱ

=X

i

P (π⁰)^∗f^∗ w_i(E)

`x(E⁰)ⁿ⁻ⁱ

so the relation x(E)ⁿ+ w1(E)x(E)ⁿ⁻¹+ ··· + wn(E)· 1 = 0 defining wi(E) pulls back to the relation x(E⁰)ⁿ+ f^∗(w₁(E))x(E⁰)ⁿ⁻¹+ ··· + f^∗(w_n(E))· 1 = 0 defining w_i(E⁰) . By the uniqueness of this relation, w_i(E⁰)= f^∗(w_i(E)) .

Proceeding to property (b), the inclusions of E₁ and E₂ into E₁⊕E₂ give in-clusions of P (E₁) and P (E₂) into P (E₁⊕E₂) with P (E₁)∩ P(E2) = ∅. Let U1 = P (E₁⊕^E2)− P(E₁) and U₂ = P(E₁⊕^E2)− P(E₂) . These are open sets in P (E₁⊕^E2) that deformation retract onto P (E₂) and P (E₁) , respectively. A map g : E₁⊕^E2

→

^R∞

which is a linear injection on fibers restricts to such a map on E₁ and E₂, so the canonical class x∈ H¹(P (E₁⊕E₂);Z2) for E₁⊕E₂ restricts to the canonical classes for E₁ and E₂. If E₁ and E₂ have dimensions m and n , consider the classes ω₁ = P

jw_j(E₁)x^m−j and ω₂ = P

jw_j(E₂)x^n−j in H^∗(P (E₁⊕^E2);Z₂) , with cup product ω₁ω₂ =P

j

P

r+s=jw_r(E₁)w_s(E₂)

x^m+n−j. By the definition of the classes w_j(E₁) , the class ω₁ restricts to zero in H^m(P (E₁);Z2) , hence ω₁ pulls back to a class in the relative group H^m(P (E₁⊕E₂), P (E₁);Z2) ≈ H^m(P (E₁⊕E₂), U₂;Z2) , and similarly for ω₂. The following commutative diagram, with Z2 coefficients understood, then shows that ω₁ω₂= 0:

−−−−−→ −−−−−→ −−−−−→

^{UEPHn 1 E2 1}

⁽

⁽

⁾

^{),⊕ UEPHm n 1 E2 1 U 02}

⁽

⁽

⁾

^),⊕

U E

P

H^m

(

( 1_⊕E2_), 2

)

_× ⁺ ∪ =

−−−−−−−−−−−−−−−→

E P

Hⁿ

(

( 1⊕E2)

)

H^{m n}

(

P(E1⊕E2)

)

E

P

H^m

(

( 1⊕E2)

)

× ⁺ Thus ω₁ω₂ =P

j

P

r+s=jw_r(E₁)w_s(E₂)

x^m+n−j = 0 is the defining relation for the Stiefel-Whitney classes of E₁⊕^E2, and so w_j(E₁⊕^E2)=P

r+s=jw_r(E₁)w_s(E₂) . Property (c) holds by definition. For (d), recall that the canonical line bundle is E = { (`, v) ∈ RP<sup>∞</sup>×R<sup>∞</sup> | v ∈ ` }. The map P(π) in this case is the identity. The map g : E

→

^R∞ which is a linear injection on fibers can be taken to be g(`, v)= v . So P (g) is also the identity, hence x(E) is a generator of H¹(RP^∞;Z₂) . The defining relation x(E)+ w1(E)· 1 = 0 then says that w1(E) is a generator of H¹(RP^∞;Z2) .

The proof of uniqueness of the classes w_i will use a general property of vector bundles called the splitting principle:

P

roposition 3.3. For each vector bundle π : E

→

B there is a space F (E) and a map p : F (E)

→

B such that the pullback p^∗(E)

→

F (E) splits as a direct sum of line bun-dles, and p^∗: H^∗(B;Z₂)

→

^{H∗(F (E);Z2}) is injective.

P

roof: Consider the pullback P (π )^∗(E) of E via the map P (π ) : P (E)

→

B . This pull-back contains a natural one-dimensional subbundle L= { (`, v) ∈ P(E)×E | v ∈ ` }.

An inner product on E pulls back to an inner product on the pullback bundle, so we have a splitting of the pullback as a sum L⊕^L⊥ with the orthogonal bundle L^⊥ hav-ing dimension one less than E . As we have seen, the Leray-Hirsch theorem applies to P (E)

→

^{B , so H∗(P (E);Z2}) is the free H^∗(B;Z2) module with basis 1, x,··· , xⁿ⁻¹

and in particular the induced map H^∗(B;Z2)

→

^{H∗(P (E);Z2}) is injective since one of the basis elements is 1 .

This construction can be repeated with L^⊥

→

P (E) in place of E

→

B . After finitely

many repetitions we obtain the desired result. tu

Looking at this construction a little more closely, L^⊥ consists of pairs (`, v) ∈ P (E)×E with v ⊥ `. At the next stage we form P(L^⊥) , whose points are pairs (`, `⁰) where ` and `⁰ are orthogonal lines in E . Continuing in this way, we see that the final base space F (E) is the space of all orthogonal splittings `<sub>1</sub>⊕ ··· ⊕ `n of fibers of E as sums of lines, and the vector bundle over F (E) consists of all n tuples of vectors in these lines. Alternatively, F (E) can be described as the space of all chains V₁ ⊂ ··· ⊂ Vn of linear subspaces of fibers of E with dim V_i = i. Such chains are called flags, and F (E)

→

B is the flag bundle associated to E . Note that the description of points of F (E) as flags does not depend on a choice of inner product in E .

Now we can finish the proof of Theorem 3.1. Property (d) determines w₁(E) for the canonical line bundle E

→

^RP∞. Property (c) then determines all the w_i’s for this bundle. Since the canonical line bundle is the universal line bundle, property (a) therefore determines the classes w_i for all line bundles. Property (b) extends this to sums of line bundles, and finally the splitting principle implies that the w_i’s are determined for all bundles.

For complex vector bundles we can use the same proof, but with Z coefficients since H^∗(CP^∞;Z) ≈ Z[α], with α now two-dimensional. The defining relation for the c_i(E) ’s is modified to be

xⁿ− c₁(E)xⁿ⁻¹+ ··· + (−1)ⁿc_n(E)· 1 = 0

with alternating signs. This is equivalent to changing the sign of α , so it does not affect the proofs of properties (a)–(c), but it has the advantage that the canonical line bundle E

→

^{CP∞ has c1(E)} = α rather than −α, since the defining relation in this

case is x(E)− c1(E)· 1 = 0 and x(E) = α. tu

Note that in property (d) for Stiefel-Whitney classes we could just as well use the canonical line bundle overRP¹instead ofRP^∞since the inclusionRP¹

>

^{RP∞ induces}

an isomorphism H¹(RP^∞;Z2)≈ H¹(RP¹;Z2) . The analogous remark for Chern classes is valid as well.

E

xample 3.4. Property (a), the naturality of Stiefel-Whitney classes, implies that a product bundle E = B×Rⁿ has w_i(E)= 0 for i > 0 since a product is the pullback of a bundle over a point, which must have w_i = 0 for i > 0 since a point has trivial cohomology in positive dimensions.

E

xample 3.5: Stability. Property (b) implies that taking the direct sum of a bundle with a product bundle does not change its Whitney classes. In this sense Stiefel-Whitney classes are stable. For example, the tangent bundle T Sⁿ to Sⁿ is stably

trivial since its direct sum with the normal bundle to Sⁿ in Rⁿ⁺¹, which is a trivial line bundle, produces a trivial bundle. Hence the Stiefel-Whitney classes w_i(T Sⁿ) are zero for i > 0 .

From the identity

(1+ w₁+ w₂+ ···)(1 + w₁⁰+ w₂⁰+ ···) = 1 + (w₁+ w₁⁰)+ (w₂+ w₁w₁⁰+ w₂⁰)+ ···

we see that w(E₁) and w(E₁⊕^E2) determine w(E₂) since the equations w₁+ w1⁰ = a1

w₂+ w1w₁⁰+ w2⁰ = a2

P···

iw_n−iw_i⁰= a_n

can be solved successively for the w_i⁰’s in terms of the w_i’s and a_i’s. In particular, if E₁⊕^E2 is the trivial bundle, then we have the case that a_i= 0 for i > 0 and so w(E1) determines w(E₂) uniquely by explicit formulas that can be worked out. For example, w₁⁰= −w1 and w₂⁰ = −w1w₁⁰− w2= w1²− w2. Of course forZ2 coefficients the signs do not matter, but the same reasoning applies to Chern classes, with Z coefficients.

E

xample 3.6. Let us illustrate this principle by showing that there is no bundle E

→

^RP∞ whose sum with the canonical line bundle E₁(R^∞) is trivial. For we have w(E₁(R^∞))= 1 + ω where ω is a generator of H¹(RP^∞;Z₂) , and hence w(E) must be (1+ ω)⁻¹= 1 + ω + ω²+ ··· since we are using Z₂ coefficients. Thus w_i(E)= ωⁱ, which is nonzero in H^∗(RP^∞;Z2) for all i . However, this contradicts the fact that w_i(E)= 0 for i > dim E .

This shows the necessity of the compactness assumption in Proposition 1.9. To further delineate the question, note that Proposition 1.9 says that the restriction E₁(Rⁿ⁺¹) of the canonical line bundle to the subspace RPⁿ ⊂ RP^∞ does have an

‘inverse’ bundle. In fact, the bundle E₁^⊥(Rⁿ⁺¹) consisting of pairs (`, v) where ` is a line through the origin in Rⁿ⁺¹ and v is a vector orthogonal to ` is such an inverse. But for any bundle E

→

^RPn whose sum with E₁(Rⁿ⁺¹) is trivial we must have w(E) = 1 + ω + ··· + ωⁿ, and since w_n(E) = ωⁿ ≠ 0, E must be at least n dimensional. So we see there is no chance of choosing such bundles E for varying n so that they fit together to form a single bundle overRP^∞.

E

xample 3.7. Let us describe an n dimensional vector bundle E

→

^{B with w}i(E) nonzero for each i≤ n. This will be the n fold Cartesian product (E1)ⁿ

→

^{(G1)n of}

the canonical line bundle over G₁= RP^∞ with itself. This vector bundle is the direct sum π₁^∗(E₁)⊕···⊕^πn^∗(E₁) where π_i: (G₁)ⁿ

→

^G1 is projection onto the i^thfactor, so w((E₁)ⁿ)=Q

i(1+ α_i)∈ Z₂[α₁,··· , α_n]≈ H^∗((RP^∞)ⁿ;Z₂) . Hence w_i((E₁)ⁿ) is the i^thelementary symmetric polynomial σ_i in the α_j’s, the sum of all the products of i different α_j’s. For example, if n= 3 then σ1= α1+α2+α3, σ₂= α1α₂+α1α₃+α2α₃,

and σ₃ = α1α₂α₃. Since each σ_i with i≤ n is nonzero in Z2[α₁,··· , αn] , we have an n dimensional bundle whose first n Stiefel-Whitney classes are all nonzero.

The same reasoning applies in the complex case to show that the n fold Cartesian product of the canonical line bundle overCP^∞ has its first n Chern classes nonzero.

In this example we see that the w_i’s and c_i’s can be identified with elementary symmetric functions, and in fact this can be done in general using the splitting princi-ple. Given an n dimensional vector bundle E

→

B we know that the pullback to F (E) splits as a sum L₁⊕···⊕L_n

→

F (E) . Letting α_i = w1(L_i) , we see that w(E) pulls back to w(L₁⊕···⊕^Ln)= (1 + α₁)··· (1 + α_n)= 1 + σ₁+ ··· + σ_n, so w_i(E) pulls back to σ_i. Thus we have embedded H^∗(B;Z2) in a larger ring H^∗(F (E);Z2) such that w_i(E) becomes the i^thelementary symmetric polynomial in the elements α₁,··· , αn

of H^∗(F (E);Z2) .

Besides the evident formal similarity between Stiefel-Whitney and Chern classes there is also a direct relation:

P

roposition 3.8. Regarding an n dimensional complex vector bundle E

→

^{B as a}

2n dimensional real vector bundle, then w_2i+1(E)= 0 and w2i(E) is the image of c_i(E) under the coefficient homomorphism H²ⁱ(B;Z)

→

^{H2i(B;Z2) .}

For example, since the canonical complex line bundle overCP^∞has c₁ a generator of H²(CP^∞;Z), the same is true for its restriction over S²= CP¹, so by the proposition this 2 dimensional real vector bundle E

→

^{S2 has w2(E)≠ 0.}

P

roof: The bundle E has two projectivizationsRP(E) and CP(E), consisting of all the real and all the complex lines in fibers of E , respectively. There is a natural projection p :RP(E)

→

CP(E) sending each real line to the complex line containing it, since a real line is all the real scalar multiples of any nonzero vector in it and a complex line is all the complex scalar multiples. This projection p fits into a commutative diagram

−−−−−→ −−−−−→ −−−−−→ −−−−−−−−−−→ −−−−−−−−−→

^{EPP2 ( )}

R R ^g

p

P ( )

R RP

-n 1 ∞

−−−−−−−−−−→

−−−−−−−−−→

^{P E}

P ( )

C ^{n 1-} C ^{CP ( )g} CP^∞

where the left column is the restriction of p to a fiber of E and the maps RP(g) and CP(g) are obtained by projectivizing, over R and C, a map g : E

→

^{C∞ which}

is a C linear injection on fibers. It is easy to see that all three vertical maps in this diagram are fiber bundles with fiber RP¹, the real lines in a complex line. The Leray-Hirsch theorem applies to the bundle RP^∞

→

^{CP∞, with Z2} coefficients, so if β is the standard generator of H²(CP^∞;Z), the Z₂ reduction β∈ H²(CP^∞;Z₂) pulls back to a generator of H²(RP^∞;Z₂) , namely the square α² of the generator α ∈ H¹(RP^∞;Z2) . Hence the Z2 reduction x_C(E) = CP(g)^∗(β) ∈ H²(CP(E); Z2) of the basic class x_C(E) = CP(g)^∗(β) pulls back to the square of the basic class x_R(E) =

RP(g)^∗(α)∈ H¹(RP(E); Z2) . Consequently the Z2 reduction of the defining relation for the Chern classes of E , which is x_C(E)ⁿ+ c₁(E)x_C(E)ⁿ⁻¹+ ··· + c_n(E)· 1 = 0, pulls back to the relation x_R(E)²ⁿ+ c₁(E)x_R(E)²ⁿ⁻²+ ··· + c_n(E)· 1 = 0, which is the defining relation for the Stiefel-Whitney classes of E . This means that w_2i+1(E)= 0

and w_2i(E)= ci(E) . tu

在文檔中 1.2. Classifying Vector Bundles (頁 78-84)