The Universal Bundle

We will show that there is a special n dimensional vector bundle E_n

→

^{Gnwith the}

property that all n dimensional bundles over paracompact base spaces are obtainable as pullbacks of this single bundle. When n= 1 this bundle will be just the canonical line bundle over RP^∞, defined earlier. The generalization to n > 1 will consist in

replacing RP^∞, the space of 1 dimensional vector subspaces ofR^∞, by the space of n dimensional vector subspaces of R^∞.

First we define the Grassmann manifold G_n(R^k) for nonnegative integers n≤ k.

As a set this is the collection of all n dimensional vector subspaces of R^k, that is, n dimensional planes in R^k passing through the origin. To define a topology on G_n(R^k) we first define the Stiefel manifold V_n(R^k) to be the space of orthonormal n frames in R^k, in other words, n tuples of orthonormal vectors in R^k. This is a subspace of the product of n copies of the unit sphere S^k−1, namely, the subspace of orthogonal n tuples. It is a closed subspace since orthogonality of two vectors can be expressed by an algebraic equation. Hence V_n(R^k) is compact since the product of spheres is compact. There is a natural surjection V_n(R^k)

→

^Gn(Rk) sending an n frame to the subspace it spans, and G_n(R^k) is topologized by giving it the quotient topology with respect to this surjection. So G_n(R^k) is compact as well. Later in this section we will construct a finite CW complex structure on G_n(R^k) and in the process show that it is Hausdorff and a manifold of dimension n(k− n).

The inclusions R^k ⊂ R^k+1 ⊂ ··· give inclusions Gn(R^k) ⊂ Gn(R^k+1)⊂ ···, and we let G_n(R^∞)=S

kG_n(R^k) . This is the set of all n dimensional vector subspaces of the vector space R^∞ since if we choose a basis for a finite-dimensional subspace of R^∞, each basis vector will lie in some R^k, hence there will be anR^k containing all the basis vectors and therefore the whole subspace. We give G_n(R^∞) the weak or direct limit topology, so a set in G_n(R^∞) is open iff it intersects each G_n(R^k) in an open set.

There are canonical n dimensional vector bundles over G_n(R^k) and G_n(R^∞) . Define E_n(R^k) = { (`, v) ∈ G<sub>n</sub>(R<sup>k</sup>)×R<sup>k</sup> | v ∈ ` }. The inclusions R^k ⊂ R^k+1 ⊂ ···

give inclusions E_n(R^k)⊂ E_n(R^k+1)⊂ ··· and we set E_n(R^∞)=S

kE_n(R^k) , again with the weak or direct limit topology.

L

emma 1.15. The projection p : E_n(R^k)

→

^Gn(Rk) , p(`, v)= `, is a vector bundle., both for finite and infinite k .

P

roof: First suppose k is finite. For `∈ G<sub>n</sub>(R<sup>k</sup>) , let π<sub>`</sub>:R^k

→

` be orthogonal projec-tion and let U<sub>`</sub>= { `<sup>0</sup>∈ G<sub>n</sub>(R<sup>k</sup>) ||π`(`<sup>0</sup>) has dimension n}. In particular, ` ∈ U_{`</sub>. We will show that U<sub>`} is open in G_n(R^k) and that the map h : p⁻¹(U_`)

→

^{U`×` ≈ U`×Rn}

defined by h(`<sup>0</sup>, v)= (`⁰, π_`(v)) is a local trivialization of E_n(R^k) .

For U_{`</sub> to be open is equivalent to its preimage in V<sub>n</sub>(R<sup>k</sup>) being open. This preimage consists of orthonormal frames v<sub>1</sub>,··· , v<sub>n</sub> such that π<sub>`}(v₁),··· , π_{`</sub>(v<sub>n</sub>) are independent. Let A be the matrix of π<sub>`} with respect to the standard basis in the domain R^k and any fixed basis in the range ` . The condition on v₁,··· , vn is then that the n×n matrix with columns Av₁,··· , Av_n have nonzero determinant.

Since the value of this determinant is obviously a continuous function of v₁,··· , v_n, it follows that the frames v₁,··· , vn yielding a nonzero determinant form an open set in V_n(R^k) .

It is clear that h is a bijection which is a linear isomorphism on each fiber. We need to check that h and h⁻¹ are continuous. For `<sup>0</sup>∈ U<sub>`</sub> there is a unique invertible linear map L_`0:R^k

→

^Rk restricting to π_{`</sub> on `⁰ and the identity on `<sup>⊥</sup>= Ker π<sub>`}. We claim that L<sub>`</sub>0, regarded as a k×k matrix, depends continuously on `⁰. Namely, we can write L_`0 as a product AB⁻¹ where:

— B sends the standard basis to v₁,··· , v_n, v_n+1,··· , v_k with v₁,··· , v_n an or-thonormal basis for `<sup>0</sup> and v<sub>n+1</sub>,··· , v<sub>k</sub> a fixed basis for `^⊥.

— A sends the standard basis to π<sub>`</sub>(v<sub>1</sub>),··· , π`(v_n), v_n+1,··· , vk.

Both A and B depend continuously on v₁,··· , vn. Since matrix multiplication and matrix inversion are continuous operations (think of the ‘classical adjoint’ formula for the inverse of a matrix), it follows that the product L_{`</sub>0 = AB<sup>−1</sup> depends continuously on v<sub>1</sub>,··· , vn. But since L<sub>`}0 depends only on `<sup>0</sup>, not on the basis v<sub>1</sub>,··· , vnfor `⁰, it follows that L<sub>`</sub>0 depends continuously on `⁰ since G_n(R^k) has the quotient topology from V_n(R^k) . Since we have h(`<sup>0</sup>, v) = (`⁰, π<sub>`</sub>(v))= (`⁰, L_{`</sub>0(v)) , we see that h is continuous. Similarly, h<sup>−1</sup>(`⁰, w) = (`<sup>0</sup>, L<sup>−1</sup><sub>`}0 (w)) and L⁻¹_`0 depends continuously on

`⁰, matrix inversion being continuous, so h⁻¹ is continuous.

This finishes the proof for finite k . When k= ∞ one takes U` to be the union of the U<sub>`</sub>’s for increasing k . The local trivializations h constructed above for finite k then fit together to give a local trivialization over this U_`, continuity being automatic

since we use the weak topology. tu

We will mainly be interested in the case k= ∞ now, and to simplify notation we will write G_n for G_n(R^∞) and E_n for E_n(R^∞) . As earlier in this section, we use the notation [X, Y ] for the set of homotopy classes of maps f : X

→

^{Y .}

T

heorem 1.16. For paracompact X , the map [X, G_n]

→

^{Vectn(X) , [f ]}

,

^{f∗(En) ,}

is a bijection.

Thus, vector bundles over a fixed base space are classified by homotopy classes of maps into G_n. Because of this, G_nis called the classifying space for n dimensional vector bundles and E_n

→

^Gn is called the universal bundle.

As an example of how a vector bundle could be isomorphic to a pullback f^∗(E_n) , consider the tangent bundle to Sⁿ. This is the vector bundle p : E

→

^{Sn where E =}

{ (x, v) ∈ Sⁿ×Rⁿ⁺¹| x ⊥ v }. Each fiber p⁻¹(x) is a point in G_n(Rⁿ⁺¹) , so we have a map Sⁿ

→

^{Gn(Rn+1) , x}

,

^p−1(x) . Via the inclusionRⁿ⁺¹

>

^R∞ we can view this as a map f : Sⁿ

→

^{Gn(R∞)= Gn}, and E is exactly the pullback f^∗(E_n) .

P

roof of 1.16: The key observation is the following: For an n dimensional vector bundle p : E

→

X , an isomorphism E≈ f^∗(E_n) is equivalent to a map g : E

→

^{R∞ that}

is a linear injection on each fiber. To see this, suppose first that we have a map f : X

→

^Gn and an isomorphism E≈ f^∗(E_n) . Then we have a commutative diagram

−−→ −−→ −−−−−→ −−−−−→

^E

E f E

−−−−−→

−−−−−→

X ^f G

f p

∗ ∼ π

( )

n n

n R

≈ ^∞

where π (`, v)= v . The composition across the top row is a map g : E

→

^{R∞ that is}

a linear injection on each fiber, since both ef and π have this property. Conversely, given a map g : E

→

^R∞ that is a linear injection on each fiber, define f : X

→

^{Gn by}

letting f (x) be the n plane g(p⁻¹(x)) . This clearly yields a commutative diagram as above.

To show surjectivity of the map [X, G_n]

-→

^Vectn(X) , suppose p : E

→

^{X is an}

n dimensional vector bundle. Let {U_α} be an open cover of X such that E is trivial over each U_α. By Lemma 1.21 in the Appendix to this chapter there is a countable open cover {Ui} of X such that E is trivial over each Ui, and there is a partition of unity {ϕi} with ϕi supported in U_i. Let g_i: p⁻¹(U_i)

→

^Rn be the composition of a trivialization p⁻¹(U_i)

→

^Ui×Rn with projection onto Rⁿ. The map (ϕ_ip)g_i, v

,

^ϕi(p(v))gi(v) , extends to a map E

→

^Rn that is zero outside p⁻¹(U_i) . Near each point of X only finitely many ϕ_i’s are nonzero, and at least one ϕ_i is nonzero, so these extended (ϕ_ip)g_i’s are the coordinates of a map g : E

→

^{(Rn)∞= R∞ that is}

a linear injection on each fiber.

For injectivity, if we have isomorphisms E ≈ f0^∗(E_n) and E ≈ f1^∗(E_n) for two maps f₀, f₁: X

→

^Gn, then these give maps g₀, g₁: E

→

^R∞ that are linear injections on fibers, as in the first paragraph of the proof. We claim g₀ and g₁ are homotopic through maps g_t that are linear injections on fibers. If this is so, then f₀ and f₁ will be homotopic via f_t(x)= gt(p⁻¹(x)) .

The first step in constructing a homotopy g_t is to compose g₀with the homotopy L_t:R^∞

→

^R∞ defined by L_t(x₁, x₂,···) = (1 − t)(x₁, x₂,···) + t(x₁, 0, x₂, 0,···). For each t this is a linear map whose kernel is easily computed to be 0 , so L_t is injective.

Composing the homotopy L_t with g₀ moves the image of g₀ into the odd-numbered coordinates. Similarly we can homotope g₁ into the even-numbered coordinates. Still calling the new g ’s g₀ and g₁, let g_t = (1 − t)g₀+ tg₁. This is linear and injective on fibers for each t since g₀ and g₁ are linear and injective on fibers. tu

Usually [X, G_n] is too difficult to compute explicitly, so this theorem is of limited use as a tool for explicitly classifying vector bundles over a given base space. Its importance is due more to its theoretical implications. Among other things, it can reduce the proof of a general statement to the special case of the universal bundle.

For example, it is easy to deduce that vector bundles over a paracompact base have inner products, since the bundle E_n

→

^Gn has an obvious inner product obtained by restricting the standard inner product inR^∞ to each n plane, and this inner product on E_n induces an inner product on every pullback f^∗(E_n) .

The preceding constructions and results hold equally well for vector bundles over C, with G_n(C^k) the space of n dimensional C linear subspaces of C^k, and so on. In particular, the proof of Theorem 1.16 translates directly to complex vector bundles, showing that Vectⁿ_C(X)≈ [X, Gn(C^∞)] .

There is also a version of Theorem ?? for oriented vector bundles. Let eG_n(R^k) be the space of oriented n planes in R^k, the quotient space of V_n(R^k) obtained by identifying two n frames when they determine the same oriented subspace of R^k. Forming the union over increasing k we then have the space eG_n(R^∞) . The universal oriented bundle eE_n(R^∞) over eG_n(R^∞) consists of pairs (`, v)∈ eG<sub>n</sub>(R<sup>∞</sup>)×R<sup>∞</sup> with v ∈ `. In other words, eE_n(R^∞) is the pullback of E_n(R^∞) via the natural projection Ge_n(R^∞)

→

^Gn(R∞) . Small modifications in the proof that Vectⁿ(X) ≈ [X, Gn(R^∞)]

show that Vectⁿ₊(X)≈ [X, eG_n(R^∞)] .

Both G_n(R^∞) and eG_n(R^∞) are path-connected since Vectⁿ(X) and Vectⁿ₊(X) have a single element when X is a point. The natural projection eG_n(R^∞)

→

^Gn(R∞)

ob-tained by ignoring orientations is two-to-one, and readers familiar with the notion of a covering space will have no trouble in recognizing that this two-to-one projection map is a covering space, using for example the local trivializations constructed in Lemma

??. In fact eG_n(R^∞) is the universal cover of G_n(R^∞) since it is simply-connected, be-cause of the triviality of Vectⁿ₊(S¹)≈ [S¹, eG_n(R^∞)] . One can also observe that a vector bundle E≈ f^∗(E_n(R^∞) is orientable iff its classifying map f : X

→

^Gn(R∞) lifts to a map ef : X

→

^Gen(R∞) , and in fact orientations of E correspond bijectively with lifts ef .

在文檔中 1.2. Classifying Vector Bundles (頁 31-35)