1.2. Classifying Vector Bundles

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In this section we give two homotopy-theoretic descriptions of Vectn.X.. The first

works for arbitrary paracompact spaces X, and is therefore of considerable theoretical

importance. The second is restricted to the case that X is a suspension, but is more

amenable to the explicit calculation of a number of simple examples, such as X . Sn

for small values of n.

The Universal Bundle

We will show that there is a special n dimensional vector bundle En!Gn with 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 RP1, defined earlier. The generalization to n > 1 will consist in

replacing RP1, the space of 1 dimensional vector subspaces of R1, by the space of

n dimensional vector subspaces of R1.

First we define the Grassmann manifold Gn.Rk. for nonnegative integers n _ k.

As a set this is the collection of all n dimensional vector subspaces of Rk , that is,

n dimensional planes in Rk passing through the origin. To define a topology on

Gn.Rk. we first define the Stiefel manifold Vn.Rk. to be the space of orthonormal

n frames in Rk , in other words, n tuples of orthonormal vectors in Rk . This is a

subspace of the product of n copies of the unit sphere Sk−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 Vn.Rk. is compact since the product

of spheres is compact. There is a natural surjection Vn.Rk.!Gn.Rk. sending an

n frame to the subspace it spans, and Gn.Rk. is topologized by giving it the quotient

topology with respect to this surjection. So Gn.Rk. is compact as well. Later in this

Classifying Vector Bundles Section 1.2 13

section we will construct a finite CW complex structure on Gn.Rk. and in the process

show that it is Hausdorff and a manifold of dimension n.k − n..

Define En.Rk. . f.`;v. 2 Gn.Rk._Rk j v 2 ` g. The inclusions Rk _ Rk.1 _ ___

give inclusions Gn.Rk. _ Gn.Rk.1. _ ___ and En.Rk. _ En.Rk.1. _ ___. We set

Gn

. Gn.R1. .

S

k Gn.Rk. and En

. En.R1. .

S

k En.Rk. with the weak, or direct

limit, topologies. Thus a set in Gn.R1. is open iff it intersects each Gn.Rk. in an

open set, and similarly for En.R1..

Lemma 1.7. The projection p : En.Rk.!Gn.Rk., p.`;v. . `, is a vector bundle.,

both for finite and infinite k.

Proof: First suppose k is finite. For ` 2 Gn.Rk., let _` :Rk!` be orthogonal projection

and let U`

. f`0 2 Gn.Rk. jj_`.`0. has dimension ng. In particular, ` 2 U`. We

will show that U` is open in Gn.Rk. and that the map h:p−1.U`.!U`

_` _ U`

_Rn

defined by h.`0; v. . .`0;_`.v.. is a local trivialization of En.Rk..

For U` to be open is equivalent to its preimage in Vn.Rk. being open. This

preimage consists of orthonormal frames v1; ___ ; vn such that _`.v1.; ___;_`.vn.

are independent. Let A be the matrix of _` with respect to the standard basis in

the domain Rk and any fixed basis in the range `. The condition on v1; ___ ; vn is

then that the n_n matrix with columns Av1; ___;Avn have nonzero determinant.

Since the value of this determinant is obviously a continuous function of v1; ___ ; vn,

it follows that the frames v1; ___ ; vn yielding a nonzero determinant form an open

set in Vn.Rk..

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 `0 2 U` there is a unique invertible

linear map L`0 :Rk!Rk restricting to _` on `0 and the identity on `? . Ker_`. We

claim that L`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; ___ ; vn; vn.1; ___ ; vk with v1; ___ ; vn an orthonormal

basis for `0 and vn.1; ___ ; vk a fixed basis for `? .

— A sends the standard basis to _`.v1.; ___;_`.vn.;vn.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`0 . AB−1 depends continuously

on v1; ___ ; vn. But since L`0 depends only on `0 , not on the basis v1; ___ ; vn for `0 , it

follows that L`0 depends continuously on `0 since Gn.Rk. has the quotient topology

from Vn.Rk.. Since we have h.`0; v. . .`0;_`.v.. . .`0; L`0.v.., we see that h is

continuous. Similarly, h−1.`0;w. . .`0; L−1

`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.1 one takes U` to be the union of

the U` ’s for increasing k. The local trivializations h constructed above for finite k

14 Chapter 1 Vector Bundles

then fit together to give a local trivialization over this U` , continuity being automatic

since we use the weak topology. tu

Let .X; Y. denote the set of homotopy classes of maps f :X!Y .

Theorem 1.8. For paracompact X, the map .X;Gn.!Vectn.X., .f .,f _.En., is

a bijection.

Thus, vector bundles over a fixed base space are classified by homotopy classes

of maps into Gn . Because of this, Gn is called the classifying space for n dimensional

vector bundles and En!Gn is called the universal bundle.

As an example of how a vector bundle could be isomorphic to a pullback f _.En.,

consider the tangent bundle to Sn . This is the vector bundle p : E!Sn where E .

f .x;v. 2 Sn_Rn.1 j x ? v g. Each fiber p−1.x. is a point in Gn.Rn.1., so we have

a map Sn!Gn.Rn.1., x,p−1.x.. Via the inclusion Rn.1>R1 we can view this

as a map f : Sn!Gn.R1. . Gn , and E is exactly the pullback f _.En..

Proof of 1.8: The key observation is the following: For an n dimensional vector

bundle p : E!X, an isomorphism E _ f _.En. is equivalent to a map g : E!R1 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 _.En.. Then we have a commutative diagram

¡!

¡!

E f E ¡¡¡¡¡!E ¡¡¡¡¡!

¡¡¡¡¡!

¡¡¡!X G f

f

p

¤ ¼

»

( )

n

¼ n n R1

where _.`;v. . v. The composition across the top row is a map g : E!R1 that is

a linear injection on each fiber, since both e f and _ have this property. Conversely,

given a map g : E!R1 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;Gn. -!Vectn.X., suppose p : E!X is an

n dimensional vector bundle. Let fU_

g be an open cover of X such that E is trivial

over each U_ . By Lemma 1.19 in the Appendix to this chapter there is a countable

open cover fUi

g of X such that E is trivial over each Ui , and there is a partition

of unity f'i

g with 'i supported in Ui . Let gi :p−1.Ui.!Rn be the composition

of a trivialization p−1.Ui.!Ui

_Rn with projection onto Rn . The map .'ip.gi ,

v ,'i.p.v..gi.v., extends to a map E!Rn that is zero outside p−1.Ui.. Near

each point of X only finitely many 'i ’s are nonzero, and at least one 'i is nonzero,

so these extended .'ip.gi ’s are the coordinates of a map g : E!.Rn.1 . R1 that is

a linear injection on each fiber.

For injectivity, if we have isomorphisms E _ f _

0 .En. and E _ f _

1 .En. for two

maps f0; f1 :X!Gn, then these give maps g0; g1 :E!R1 that are linear injections

Classifying Vector Bundles Section 1.2 15

on fibers, as in the first paragraph of the proof. We claim g0 and g1 are homotopic

through maps gt that are linear injections on fibers. If this is so, then f0 and f1 will

be homotopic via ft.x. . gt.p−1.x...

The first step in constructing a homotopy gt is to compose g0 with the homotopy

Lt :R1!R1 defined by Lt.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 Lt is injective.

Composing the homotopy Lt 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 gt

. .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;Gn. 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 En!Gn has an obvious inner product obtained by

restricting the standard inner product in R1 to each n plane, and this inner product

on En induces an inner product on every pullback f _.En..

The proof of the following result provides another illustration of this principle of

the ‘universal example:’

Proposition 1.9. For each vector bundle E!X with X compact Hausdorff there

exists a vector bundle E0!X such that E_E0 is the trivial bundle.

This can fail when X is noncompact. An example is the canonical line bundle

over RP1, as we shall see in Example 3.6. There are some noncompact spaces for

which the proposition remains valid, however. Among these are all infinite but finitedimensional

CW complexes, according to an exercise at the end of the chapter.

Proof: First we show this holds for En.Rk.. In this case the bundle with the desired

property will be E?

n.Rk. . f.`;v. 2 Gn.Rk._Rk j v ? ` g. This is because En.Rk. is

by its definition a subbundle of the product bundle Gn.Rk._Rk , and the construction

of a complementary orthogonal subbundle given in the proof of Proposition 1.6 yields

exactly E?

n.Rk..

Now for the general case. Let f :X!Gn pull the universal bundle En back to the

given bundle E!X. The space Gn is the union of the subspaces Gn.Rk. for k _ 1,

with the weak topology, so the following lemma implies that the compact set f.X.

must lie in Gn.Rk. for some k. Then f pulls the trivial bundle En.Rk._E?

n.Rk. back

to E_f _.E?

n.Rk.., which is therefore also trivial. tu

16 Chapter 1 Vector Bundles

Lemma 1.10. If X is the union of a sequence of subspaces X1 _ X2 _ ___ with the

weak topology, and points are closed subspaces in each Xi , then for each compact

set C _ X there is an Xi that contains C .

Proof: If the conclusion is false, then for each i there is a point xi

2 C not in Xi . Let

S . fx1;x2; ___g, an infinite set. However, S \ Xi is finite for each i, hence closed in

Xi . Since X has the weak topology, S is closed in X. By the same reasoning, every

subset of S is closed, so S has the discrete topology. Since S is a closed subspace of

the compact space C , it is compact. Hence S must be finite, a contradiction. tu

The constructions and results in this subsection hold equally well for vector bundles

over C, with Gn.Ck. the space of n dimensional C linear subspaces of Ck , etc.

In particular, the proof of Theorem 1.8 translates directly to complex vector bundles,

showing that VectnC

.X. _ .X;Gn.C1...

Vector Bundles over Spheres

Vector bundles with base space a sphere can be described more explicitly, and

this will allow us to compute Vectn.Sk. for small values of k.

First let us describe a way to construct vector bundles E!Sk . Write Sk as the

union of its upper and lower hemispheres Dk. and Dk−

, with Dk. \Dk−

. Sk−1 . Given a

map f : Sk−1!GLn.R., let Ef be the quotient of the disjoint union Dk. _RnqDk−

_Rn

obtained by identifying .x;v. 2 @Dk._Rn with .x; f .x..v.. 2 @Dk−

_Rn . There is

then a natural projection Ef!Sk and we will leave to the reader the easy verification

that this is an n dimensional vector bundle. The map f is called its clutching function.

(Presumably the terminology comes from the clutch which engages and disengages

gears in machinery.) The same construction works equally well with C in place of R,

so from a map f : Sk−1!GLn.C. one obtains a complex vector bundle Ef!Sk .

Example 1.11. Let us see how the tangent bundle TS2 to S2 can be described in these

terms. Define two orthogonal vector fields v. and w. on the northern hemisphere

D2. of S2 in the following way. Start with a standard pair of orthogonal vectors at

each point of a flat disk D2 as in the left-hand figure below, then stretch the disk over

the northern hemisphere of S2 , carrying the vectors along as tangent vectors to the

resulting curved disk. As we travel around the equator of S2 the vectors v. and w.

then rotate through an angle of 2_ relative to the equatorial direction, as in the right

half of the figure.

Classifying Vector Bundles Section 1.2 17

Reflecting everything across the equatorial plane, we obtain orthogonal vector fields

v− and w− on the southern hemisphere D2−

. The restrictions of v− and w− to the

equator also rotate through an angle of 2_ , but in the opposite direction from v.

and w. since we have reflected across the equator. The pair .v_;w_. defines a

trivialization of TS2 over D2_ taking .v_;w_. to the standard basis for R2 . Over the

equator S1 we then have two trivializations, and the function f : S1!GL2.R. which

rotates .v.;w.. to .v−;w−. sends _ 2 S1 , regarded as an angle, to rotation through

the angle 2_. For this map f we then have Ef

. TS2 .

Example 1.12. Let us find a clutching function for the canonical complex line bundle

over CP1 . S2 . (This example will play a crucial role in the next chapter.) The space

CP1 is the quotient of C2 − f0g under the equivalence relation .z0; z1. _ _.z0; z1..

Denote the equivalence class of .z0; z1. by .z0; z1.. We can also write points of CP1

as ratios z . z1=z0 2 C [ f1g . S2 . Points in the disk D2−

inside the unit circle

S1 _ C can be expressed uniquely in the form .1; z1=z0. . .1; z. with jzj _ 1, and

points in the disk D2. outside S1 can be written uniquely in the form .z0=z1; 1. .

.z−1; 1. with jz−1j _ 1. Over D2−

a section of the canonical line bundle is then given

by .1; z1=z0.,.1; z1=z0. and over D2. a section is .z0=z1; 1.,.z0=z1; 1.. These

sections determine trivializations of the canonical line bundle over these two disks,

and over their common boundary S1 we pass from the D2. trivialization to the D2−

trivialization by multiplying by z . z1=z0 . Thus the canonical line bundle is Ef for

the clutching function f : S1!GL1.C. defined by f.z. . .z..

A basic property of the construction of bundles Ef!Sk via clutching functions is

that Ef

_ Eg if f ' g. For if F : Sk−1_I!GLn.R. is a homotopy from f to g, then we

can construct by the same method a vector bundle EF!Sk_I restricting to Ef over

Sk_f0g and Eg over Sk_f1g. Hence Ef and Eg are isomorphic by Proposition 1.3.

Thus the association f,Ef gives a well-defined map Ø :_k−1GLn.R. -!Vectn.Sk.. If

we change coordinates in Rn via a fixed _ 2 GLn.R. we obtain an isomorphic bundle

E_−1f_ . Hence Ø induces a well-defined map on the set of orbits in _k−1GLn.R. under

the conjugation action of GLn.R., or what amounts to the same thing, the conjugation

action of _0GLn.R.. Since _0GLn.R. _ Z2 as we shall see below, we may write this

set of orbits as _k−1GLn.R.=Z2 .

Proposition 1.13. The map Ø :_k−1GLn.R.=Z2!Vectn.Sk. is a bijection.

Proof: An inverse mapping Ù can be constructed as follows. Given an n dimensional

vector bundle p : E!Sk , its restrictions E. and E− over Dk. and Dk−

are trivial since

Dk. and Dk−are contractible. Choose trivializations h_ : E_!Dk_ _Rn . Selecting a

basepoint s0 2 Sk−1 and fixing an isomorphism p−1.s0. _ Rn , we may assume h.

and h− are normalized to agree with this isomorphism on p−1.s0.. Then h−h−1

. defines

a map .Sk−1; s0.!.GLn.R.; 11., whose homotopy class is by definition Ù.E. 2

18 Chapter 1 Vector Bundles

_k−1GLn.R.. To see that Ù.E. is well-defined in the orbit set _k−1GLn.R.=Z2 , note

first that any two choices of normalized h_ differ by a map .Dk_; s0.!.GLn.R.; 11..

Since Dk_ is contractible, such a map is homotopic to the constant map, so the two

choices of h_ are homotopic, staying fixed over s0 . Rechoosing the identification

p−1.s0. _ Rn has the effect of conjugating Ù.E. by an element of GLn.R., so

Ù: Vectn.Sk.!_k−1GLn.R.=Z2 is well-defined.

It is clear that Ù and Ø are inverses of each other. tu

The case of complex vector bundles is similar but simpler since _0GLn.C. . 0,

and so we obtain bijections VectnC

.Sk. _ _k−1GLn.C..

The same proof shows more generally that for a suspension SX with X paracompact,

Vectn.SX. _ hX;GLn.R.i=Z2 , where hX;GLn.R.i denotes the basepointpreserving

homotopy classes of maps X!GLn.R.. In the complex case we have

VectnC

.SX. _ hX;GLn.C.i.

It is possible to compute a few homotopy groups of GLn.R. and GLn.C. by

elementary means. The first observation is that GLn.R. deformation retracts onto the

subgroup O.n. consisting of orthogonal matrices, the matrices whose columns form

an orthonormal basis for Rn , or equivalently the matrices of isometries of Rn which

fix the origin. The Gram-Schmidt process for converting a basis into an orthonormal

basis provides a retraction of GLn.R. onto O.n., continuity being evident from the

explicit formulas for the Gram-Schmidt process. Each step of the process is in fact

realizable by a homotopy, by inserting appropriate scalar factors into the formulas,

and this yields a deformation retraction of GLn.R. onto O.n.. (Alternatively, one

can use the so-called polar decomposition of matrices to show that GLn.R. is in fact

homeomorphic to the product of O.n. with a Euclidean space.) The same reasoning

shows that GLn.C. deformation retracts onto the unitary subgroup U.n., consisting

of matrices whose columns form an orthonormal basis for Cn with respect to the

standard hermitian inner product. These are the isometries in GLn.C..

Next, there are fiber bundles

O.n − 1.!- O.n. --!-p Sn−1 U.n − 1.!- U.n. --!-p S2n−1

where p is the map obtained by evaluating an isometry at a chosen unit vector, for

example .1; 0; ___ ; 0.. Local triviality for the first bundle can be shown as follows.

We can view O.n. as the Stiefel manifold Vn.Rn. by regarding the columns of an

orthogonal matrix as an orthonormal n frame. In these terms, the map p projects

an n frame onto its first vector. Given a vector v1 2 Sn−1 , extend this to an orthonormal

n frame v1; ___ ; vn. For unit vectors v near v1 , applying Gram-Schmidt

to v;v2; ___ ; vn produces a continuous family of orthonormal n frames with first vector

v . The last n−1 vectors of these frames form orthonormal bases for v? varying

continuously with v . Each such basis gives an identification of v? with Rn−1 , hence

Classifying Vector Bundles Section 1.2 19

p−1.v. is identified with Vn−1.Rn−1. . O.n − 1., and this gives the desired local

trivialization. The same argument works in the unitary case.

From the long exact sequences of homotopy groups for these bundles we deduce

immediately:

Proposition 1.14. The map _iO.n.!_iO.n.1. induced by the inclusion of O.n.

into O.n . 1. is an isomorphism for i < n−1 and a surjection for i . n − 1.

Similarly, the inclusion U.n.>U.n.1. induces an isomorphism on _i for i < 2n

and a surjection for i . 2n. tu

Here are tables of some low-dimensional calculations:

_iO.n.

n -!

1 2 3 4

i 0 Z2 Z2 Z2 Z2 ___

# 1 0 Z Z2 Z2 ___

2 0 0 0 0 ___

3 0 0 Z Z_Z

_i U

.n.

n -!

1 2 3 4

i 0 0 0 0 0 ___

# 1 Z Z Z Z ___

2 0 0 0 0 ___

3 0 Z Z Z ___

Proposition 1.14 says that along each row in the first table the groups stabilize once

we pass the diagonal term _nO.n.1., and in the second table the rows stabilize even

sooner. The stable groups are given by the famous Bott Periodicity Theorem which

we prove in Chapter 2 in the complex case and Chapter 4 in the real case:

i mod 8 0 1 2 3 4 5 6 7

_iO.n. Z2 Z2 0 Z 0 0 0 Z

_iU.n. 0 Z 0 Z 0 Z 0 Z

The calculations in the first two tables can be obtained from the following homeomorphisms,

together with the fact that the universal cover of RP3 is S3 :

O.n. _ S0_SO.n.

SO.1. . f1g

SO.2. _ S1

SO.3. _ RP3

SO.4. _ RP3_S3

U.n. _ S1_SU.n.

SU.1. . f1g

SU.2. _ S3

Here SO.n. and SU.n. are the subgroups consisting of matrices of determinant 1.

A homeomorphism O.n.!S0_SO.n. can be defined by _,.det._.;_0. where _0

is obtained from _ by multiplying its last column by the scalar 1= det._.. The inverse

homeomorphism sends ._; _. 2 S0_SO.n. to the matrix obtained by multiplying the

last column of _ by _. The same formulas in the complex case give a homeomorphism

U.n. _ S1_SU.n..

It is obvious that SO.1. and SU.1. are trivial. For the homeomorphisms SO.2. _

S1 and SU.2. _ S3 , note that 2_2 orthogonal or unitary matrices of determinant 1

are determined by their first column, which can be any unit vector in R2 or C2 .

20 Chapter 1 Vector Bundles

A homeomorphism SO.3. _ RP3 can be obtained in the following way. Let

':D3!SO.3. send a nonzero vector x 2 D3 to the rotation through angle jxj_

about the line determined by x. An orientation convention, such as the ‘right-hand

rule,’ is needed to make this unambiguous. By continuity, ' must send 0 to the

identity. Antipodal points of S2 . @D3 are sent to the same rotation through angle

_ , so ' induces a map ':RP3!SO.3., where RP3 is viewed as D3 with antipodal

boundary points identified. The map ' is clearly injective since the axis of a nontrivial

rotation is uniquely determined as its fixed point set, and ' is surjective since by

easy linear algebra each nonidentity element of SO.3. is a rotation about a unique

axis. It follows that ' is a homeomorphism RP3 _ SO.3..

It remains to show that SO.4. is homeomorphic to S3_SO.3.. Identifying R4

with the quaternions H and S3 with the group of unit quaternions, the quaternion

multiplication w,vw for fixed v 2 S3 defines an isometry _v

2 O.4. since quaternionic

multiplication satisfies jvwj . jvjjwj and we are taking v to be a unit vector.

Points of O.4. can be viewed as 4 tuples .v1; ___ ; v4. of orthonormal vectors

vi

2 H . R4 , and O.3. can be viewed as the subspace with v1 . 1. Define a map

S3_O.3.!O.4. by sending .v; .1; v2; v3; v4.. to .v;vv2; vv3; vv4., the result of

applying _v to the orthonormal frame .1; v2; v3; v4.. This map is a homeomorphism

since it has an inverse defined by .v;v2; v3; v4.,.v; .1; v−1v2; v−1v3; v−1v4.., the

second coordinate being the orthonormal frame obtained by applying _v−1 to the

frame .v;v2; v3; v4.. Since the path-components of S3_O.3. and O.4. are homeomorphic

to S3_SO.3. and SO.4. respectively, it follows that these path-components

are homeomorphic.

The conjugation action of _0O.n. _ Z2 on _iO.n. which appears in the bijection

Vectn.Si.1. _ _iO.n.=Z2 is trivial in the stable range i < n−1 since we

can realize each element of _iO.n. by a map Si!O.i . 1. and then act on this by

conjugating by a reflection across a hyperplane containing Ri.1 . Note that the map

Vectn.Si.1.!Vectn.1.Si.1. corresponding to the map _iO.n.!_iO.n.1. induced

by the inclusion O.n.>O.n.1. is just direct sum with the trivial line bundle. Thus

the stable isomorphism classes of vector bundles over spheres form groups, the same

groups appearing in Bott Periodicity. This is the beginning of K–theory, as we shall

see in the next chapter.

Outside the stable range the conjugation action is not always trivial. For example,

in _1O.2. _ Z the action is given by the nontrivial automorphism of Z, multiplication

by −1, since conjugating a rotation of R2 by a reflection produces a rotation in

the opposite direction. Thus 2 dimensional vector bundles over S2 are classified by

non-negative integers. When we stabilize by taking direct sum with a line bundle, then

we are in the stable range where _1O.n. _ Z2 , so the 2 dimensional bundles corresponding

to even integers are the ones which are stably trivial. The tangent bundle

Classifying Vector Bundles Section 1.2 21

T.S2. is stably trivial, hence corresponds to an even integer, in fact to 2 as we saw in

Example 2.11.

Another case in which the conjugation action on _iO.n. is trivial is when n is

odd since in this case we can choose the conjugating element to be the orientationreversing

isometry x,−x, which commutes with every linear map.

The two identifications of Vectn.Sk. with .Sk;Gn.R1.. and _k−1O.n.=Z2 are

related in the following way. First, there is a fiber bundle O.n.!Vn.R1.!Gn.R1.

where the map Vn!Gn projects an n frame onto the n plane it spans. Local triviality

follows from local triviality of the universal bundle En!Gn since Vn can be viewed

as the bundle of n frames in fibers of En . The space Vn.R1. is contractible. This can

be seen by using the embeddings Lt :R1!R1 defined in the proof of Theorem 1.8 to

deform an arbitrary n frame into the odd-numbered coordinates of R1, then taking

the standard linear deformation to a fixed n frame in the even coordinates; these

deformations may produce nonorthonormal n frames, but orthonormality can always

be restored by the Gram-Schmidt process. Since the homotopy groups of the total

space of the fiber bundle O.n.!Vn.R1.!Gn.R1. are trivial, we get isomorphisms

_kGn.R1. _ _k−1O.n.. By Proposition 4A.1 of [AT], .Sk;Gn.R1.. is _kGn.R1.

modulo the action of _1Gn.R1.. Thus Vectn.Sk. is equal to both _kGn.R1. modulo

the action of _1Gn.R1. and _k−1O.n. modulo the action of _0O.n.. One can check

that under the isomorphisms _kGn.R1. _ _k−1O.n. and _0O.n. _ _1Gn.R1. the

actions correspond, so the two descriptions of Vectn.Sk. are equivalent.

Orientable Vector Bundles

An orientation of Rn is an equivalence class of ordered bases, two ordered bases

being equivalent if the linear isomorphism taking one to the other has positive determinant.

An orientation of an n dimensional vector bundle is a choice of orientation

in each fiber which is locally constant, in the sense that it is defined in a neighborhood

of any fiber by n independent local sections.

Let Vectn

..B. be the set of orientation-preserving isomorphism classes of oriented

n dimensional vector bundles over B. The proof of Theorem 1.8 extends without

difficulty to show that Vectn

..B. _ .B; eG

n. where eG

n is the space of oriented n planes

in R1. This is the orbit space of Vn.R1 under the action of SO.n., just as Gn is

the orbit space under the action of O.n.. The universal oriented bundle e En over eG

n

consists of pairs .`;v. 2 eG

n

_R1 with v 2 `. In other words, e En!eG

n is the pullback

of En!Gn via the natural projection eG

n!Gn . It is easy to see that this projection is a

2 sheeted covering space, and an n dimensional vector bundle E!B is orientable iff

its classifying map f : B!Gn with f _.En. _ E lifts to a map e f : B!eG

n . In fact, each

lift e f corresponds to an orientation of E. The space eG

n is path-connected, since Gn is

connected and two points of eG

n having the same image in Gn are oppositely oriented

n planes which can be joined by a path in eG

n rotating the n plane 180 degrees in an

22 Chapter 1 Vector Bundles

ambient .n . 1. plane, reversing its orientation. Since _1.Gn. _ _0O.n. _ Z2 , this

implies that eG

n is the universal cover of Gn .

The oriented version of Proposition 1.13 is a bijection _k−1SO.n. _ Vectn

..Sk.,

proved in the same way. Since _0SO.n. . 0, there is no action to factor out.

Complex vector bundles are always orientable, when regarded as real vector bundles

by restricting the scalar multiplication to R. For if v1; ___ ; vn is a basis for Cn

then the basis v1; iv1; ___ ; vn; ivn for Cn as an R vector space determines an orientation

of Cn which is independent of the choice of C basis v1; ___ ; vn since any other

C basis can be joined to this one by a continuous path of C bases, the group GLn.C.

being path-connected.

A Cell Structure on Grassmann Manifolds

Since Grassmann manifolds play such a fundamental role in vector bundle theory,

it would be good to have a better grasp on their topology. Here we show that Gn.R1.

has the structure of a CW complex with each Gn.Rk. a finite subcomplex. We will

also see that Gn.Rk. is a closed manifold of dimension n.k−n.. Similar statements

hold in the complex case as well.

For a start let us show that Gn.Rk. is Hausdorff, since we will need this fact later

when we construct the CW structure. Given two n planes ` and `0 in Gn.Rk., it

suffices to find a continuous f :Gn.Rk.!R taking different values on ` and `0 . For

a vector v 2 Rk let fv.`. be the length of the orthogonal projection of v onto `.

This is a continuous function of ` since if we choose an orthonormal basis v1; ___ ; vn

for ` then fv.`. .

􀀀

.v _ v1.2 . ___..v _ vn.2_1=2 , which is certainly continuous in

v1; ___ ; vn hence in ` since Gn.Rk. has the quotient topology from Vn.Rk.. Now for

an n plane `0 . ` choose v 2 ` − `0 , and then fv.`. . jvj > fv.`0..

In order to construct the CW structure we need some notation and terminology.

In R1 we have the standard subspaces R1 _ R2 _ ___. For an n plane ` 2 Gn there

is then the increasing chain of subspaces `j

. ` \ Rj , with `j

. ` for large j . Each

`j either equals `j−1 or has dimension one greater than `j−1 since `j is spanned by

`j−1 together with any vector in `j

− `j−1 . Let _i.`. be the minimum j such that

`j has dimension i. The increasing sequence _.`. . ._1.`.; ___;_n.`.. is called the

Schubert symbol of `. For example, if ` is the standard Rn _ R1 then `j

. Rj for

j _ n and _.Rn. . .1;2; ___;n.. Clearly, Rn is the only n plane with this Schubert

symbol.

For a Schubert symbol _ . ._1; ___;_n. let e._. . f` 2 Gn

j _.`. . _ g.

Proposition 1.15. e._. is an open cell of dimension ._1−1..._2−2..___.._n

−n.,

and these cells e._. are the cells of a CW structure on Gn . The subspace Gn.Rk. is

the finite subcomplex consisting of cells with _n

_ k.

Classifying Vector Bundles Section 1.2 23

For example G2.R4. has six cells corresponding to the Schubert symbols .1; 2.,

.1; 3., .1; 4., .2; 3., .2; 4., .3; 4., and these cells have dimensions 0, 1, 2, 2, 3, 4

respectively.

Proof: Our main task will be to find a characteristic map for e._.. Note first that

e._. _ Gn.Rk. for k _ _n . Let Hi be the hemisphere in S_i−1 _ R_i _ Rk consisting

of unit vectors with non-negative _i th coordinate. In the Stiefel manifold Vn.Rk.

let E._. be the subspace of orthonormal frames .v1; ___ ; vn. 2 .Sk−1.n such that

vi

2 Hi for each i. We claim that the projection _ : E._.!H1 , _.v1; ___ ; vn. . v1 ,

is a trivial fiber bundle. This is equivalent to finding a projection p : E._.!_−1.v0.

which is a homeomorphism on fibers of _ , where v0 . .0; ___ ; 0; 1. 2 R_1 _ Rk , since

the map __p : E._.!H1__−1.v0. is then a continuous bijection of compact Hausdorff

spaces, hence a homeomorphism. The map p :_−1.v.!_−1.v0. is obtained by

applying the rotation _v of Rk that takes v to v0 and fixes the .k − 2. dimensional

subspace orthogonal to v and v0 . This rotation takes Hi to itself for i > 1 since it

affects only the first _1 coordinates of vectors in Rk . Hence p takes _−1.v. onto

_−1.v0..

The fiber _−1.v0. can be identified with E._0. for _0 . ._2 −1; ___;_n

−1.. By

induction on n this is homeomorphic to a closed ball of dimension ._2 − 2. . ___.

._n

− n., so E._. is a closed ball of dimension ._1 − 1. . ___.._n

− n..

The natural map E._.!Gn sending an orthonormal n tuple to the n plane it

spans takes the interior of the ball E._. to e._. bijectively since each ` 2 e._.

has a unique basis .v1; ___ ; vn. 2 int E._.. Namely, consider the sequence of subspaces

`_1

_ ___ _ `_n , and choose vi

2 `_i to be the unit vector with positive

_i th coordinate orthogonal to `_i−1 . Since Gn has the quotient topology from Vn ,

the map int E._.!e._. is a homeomorphism, so e._. is an open cell of dimension

._1−1..___.._n

−n.. The boundary of E._. maps to cells e._0. of Gn where _0 is

obtained from _ by decreasing some _i ’s, so these cells e._0. have lower dimension

than e._..

It is clear from the definitions that Gn.Rk. is the union of the cells e._. with

_n

_ k. To see that the maps E._.!Gn.Rk. for these cells are the characteristic

maps for a CW structure on Gn.Rk. we can argue as follows. For fixed k, let Xi

be the union of the cells e._. in Gn.Rk. having dimension at most i. Suppose by

induction on i that Xi is a CW complex with these cells. Attaching the .i . 1. cells

e._. of Xi.1 to Xi via the maps @E._.!Xi produces a CW complex Y and a natural

continuous bijection Y!Xi.1 . Since Y is a finite CW complex it is compact, and Xi.1

is Hausdorff as a subspace of Gn.Rk., so the map Y!Xi.1 is a homeomorphism

and Xi.1 is a CW complex, finishing the induction. Thus we have a CW structure on

Gn.Rk..

Since the inclusions Gn.Rk. _ Gn.Rk.1. for varying k are inclusions of subcom24

Chapter 1 Vector Bundles

plexes, and Gn.R1. has the weak topology with respect to these subspaces, it follows

that we have a CW structure on Gn.R1.. tu

Similar constructions work to give CW structures on complex Grassmann manifolds,

but here e._. will be a cell of dimension .2_1−2...2_2−4..___..2_n

−2n..

The ‘hemisphere’ Hi is defined to be the subspace of the unit sphere S2_i−1 in C_i

consisting of vectors whose _i th coordinate is non-negative real, so Hi is a ball of

dimension 2_i

−2. The transformation _v

2 SU.k. is uniquely determined by specifying

that it takes v to v0 and fixes the orthogonal .k − 2. dimensional complex

subspace, since an element of U.2. of determinant 1 is determined by where it sends

one unit vector.

The highest-dimensional cell of Gn.Rk. is e._. for _ . .k−n.1; k−n.2; ___ ; k.,

of dimension n.k−n., so this is the dimension of Gn.Rk.. Near points in these topdimensional

cells Gn.Rk. is a manifold. But Gn.Rk. is homogeneous in the sense that

given any two points in Gn.Rk. there is a homeomorphism Gn.Rk.!Gn.Rk. taking

one point to the other, namely, the homeomorphism induced by an invertible linear

map Rk!Rk taking one n plane to the other. From this homogeneity it follows that

Gn.Rk. is a manifold near all points. Since it is compact, it is a closed manifold.

There is a natural inclusion i :Gn>Gn.1 , i.`. . R_j.`. where j :R1!R1 is

the embedding j.x1;x2; ___. . .0;x1;x2; ___.. If _.`. . ._1; ___;_n. then _.i.`.. .

.1;_1.1; ___;_n

.1., so i takes cells of Gn to cells of Gn.1 of the same dimension,

making i.Gn. a subcomplex of Gn.1 . Identifying Gn with the subcomplex i.Gn.,

we obtain an increasing sequence of CW complexes G1 _ G2 _ ___ whose union

G1 .

S

n Gn is therefore also a CW complex. Similar remarks apply as well in the

complex case.

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