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Allen Hatcher

Copyright c 2003 by Allen Hatcher

Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author.

All other rights reserved.

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Introduction

. . . 1

Chapter 1. Vector Bundles

. . . 4

1.1. Basic Definitions and Constructions

. . . 6 Sections 8. Direct Sums 10. Inner Products 12. Tensor Products 14.

Associated Fiber Bundles 16.

1.2. Classifying Vector Bundles

. . . 18 Pullback Bundles 18. Clutching Functions 22. The Universal Bundle 27.

Cell Structures on Grassmannians 31. Appendix: Paracompactness 35.

Chapter 2. K–Theory

. . . 38

2.1. The Functor K(X)

. . . 39 Ring Structure 40. The Fundamental Product Theorem 41.

2.2. Bott Periodicity

. . . 51 Exact Sequences 51. Deducing Periodicity from the Product Theorem 53.

Extending to a Cohomology Theory 54. Elementary Applications 58.

2.3. Division Algebras and Parallelizable Spheres

. . . 59 H–Spaces 59. Adams Operations 62. The Splitting Principle 65.

2.4. Bott Periodicity in the Real Case

[not yet written]

2.5. Vector Fields on Spheres

[not yet written]

Chapter 3. Characteristic Classes

. . . 73

3.1. Stiefel-Whitney and Chern Classes

. . . 74 Axioms and Constructions 74. Cohomology of Grassmannians 80.

3.2. Euler and Pontryagin Classes

. . . 84 The Euler Class 87. Pontryagin Classes 90.

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4.1. Lower Bounds on Im J

. . . 95 The Chern Character 96. The e Invariant 98. Thom Spaces 99.

Bernouilli Denominators 102.

4.2. Upper Bounds on Im J

[not yet written]

Bibliography

. . . 106

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Topological K–theory, the first generalized cohomology theory to be studied thor- oughly, was introduced around 1960 by Atiyah and Hirzebruch, based on the Periodic- ity Theorem of Bott proved just a few years earlier. In some respects K–theory is more elementary than classical homology and cohomology, and it is also more powerful for certain purposes. Some of the best-known applications of algebraic topology in the twentieth century, such as the theorem of Bott and Milnor that there are no division algebras after the Cayley octonions, or Adams’ theorem determining the maximum number of linearly independent tangent vector fields on a sphere of arbitrary dimen- sion, have relatively elementary proofs using K–theory, much simpler than the original proofs using ordinary homology and cohomology.

The first portion of this book takes these theorems as its goals, with an exposition that should be accessible to bright undergraduates familiar with standard material in linear algebra, abstract algebra, and point-set topology. Later chapters of the book assume more, approximately the contents of a standard graduate course in algebraic topology. A concrete goal of the later chapters is to tell the full story on the stable J–homomorphism, which gives the first level of depth in the stable homotopy groups of spheres. Along the way various other topics related to vector bundles that are of interest independent of K–theory are also developed, such as the characteristic classes associated to the names Stiefel and Whitney, Chern, and Pontryagin.

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Everyone is familiar with the M¨obius band, the twisted product of a circle and a line, as contrasted with an annulus which is the actual product of a circle and a line.

Vector bundles are the natural generalization of the M¨obius band and annulus, with the circle replaced by an arbitrary topological space, called the base space of the vector bundle, and the line replaced by a vector space of arbitrary finite dimension, called the fiber of the vector bundle. Vector bundles thus combine topology with linear algebra, and the study of vector bundles could be called Linear Algebraic Topology.

The only two vector bundles with base space a circle and one-dimensional fiber are the M¨obius band and the annulus, but the classification of all the different vector bundles over a given base space with fiber of a given dimension is quite difficult in general. For example, when the base space is a high-dimensional sphere and the dimension of the fiber is at least three, then the classification is of the same order of difficulty as the fundamental but still largely unsolved problem of computing the homotopy groups of spheres.

In the absence of a full classification of all the different vector bundles over a given base space, there are two directions one can take to make some partial progress on the problem. One can either look for invariants to distinguish at least some of the different vector bundles, or one can look for a cruder classification, using a weaker equivalence relation than the natural notion of isomorphism for vector bundles. As it happens, the latter approach is more elementary in terms of prerequisites, so let us discuss this first.

There is a natural direct sum operation for vector bundles over a fixed base space X , which in each fiber reduces just to direct sum of vector spaces. Using this, one can obtain a weaker notion of isomorphism of vector bundles by defining two vector bun- dles over the same base space X to be stably isomorphic if they become isomorphic after direct sum with product vector bundles X×Rn for some n , perhaps different n ’s for the two given vector bundles. Then it turns out that the set of stable isomor- phism classes of vector bundles over X forms an abelian group under the direct sum operation, at least if X is compact Hausdorff. The traditional notation for this group is gKO(X) . In the case of spheres the groups gKO(Sn) have the quite unexpected prop- erty of being periodic in n . This is called Bott Periodicity, and the values of gKO(Sn) are given by the following table:

n mod 8 1 2 3 4 5 6 7 8

KO(Sg n) Z2 Z2 0 Z 0 0 0 Z

For example, gKO(S1) isZ2, a cyclic group of order two, and a generator for this group is the M¨obius bundle. This has order two since the direct sum of two copies of the

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obius bundle is the product S1×R2, as one can see by embedding two M¨obius bands in a solid torus so that they intersect orthogonally along the common core circle of both bands, which is also the core circle of the solid torus.

Things become simpler if one passes from the real vector spaces to complex vector spaces. The complex version of gKO(X) , called eK(X) , is constructed in the same way as gKO(X) but using vector bundles whose fibers are vector spaces overC rather than R. The complex form of Bott Periodicity asserts simply that eK(Sn) is Z for n even and 0 for n odd, so the period is two rather than eight.

The groups eK(X) and gKO(X) for varying X share certain formal properties with the cohomology groups studied in classical algebraic topology. Using a more general form of Bott periodicity, it is in fact possible to extend the groups eK(X) and gKO(X) to a full cohomology theory, families of abelian groups eKn(X) and gKOn(X) for n∈ Z that are periodic in n of period two and eight, respectively. There is more algebraic structure here than just the additive group structure, however. Tensor products of vector spaces give rise to tensor products of vector bundles, which in turn give prod- uct operations in both real and complex K–theory similar to cup product in ordinary cohomology. Furthermore, exterior powers of vector spaces give natural operations within K–theory.

With all this extra structure, K–theory becomes a powerful tool, in some ways more powerful even than ordinary cohomology. The prime example of this is the very simple proof, once the basic machinery of complex K–theory has been set up, of the theorem that there are no finite dimensional division algebras over R in dimensions other than 1 , 2 , 4 , and 8 , the dimensions of the classical examples of the real and complex numbers, the quaternions, and the Cayley octonions. The same proof shows also that the only spheres whose tangent bundles are product bundles are S1, S3, and S7, the unit spheres in the complex numbers, quaternions, and octonions.

Another classical problem that can be solved more easily using K–theory than ordinary cohomology is to find the maximum number of linearly independent tangent vector fields on the sphere Sn. In this case complex K–theory is not enough, and the added subtlety of real K–theory is needed. There is an algebraic construction of the requisite number of vector fields using Clifford algebras, and the harder part is to show there can be no more than this construction provides. Clifford algebras also provide a nice explanation for the mysterious sequence of groups appearing the in real form of Bott periodicity.

Now let us return to the original classification problem for vector bundles over a given base space and the question of finding invariants to distinguish different vector bundles. The first such invariant is orientability, the question of whether all the fibers can be coherently oriented. For example, the M¨obius bundle is not orientable since as one goes all the way around the base circle, the orientation of the fiber lines is reversed. This does not happen for the annulus, which is an orientable vector bundle.

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Orientability is measured by the first of a sequence of cohomology classes associ- ated to a vector bundle, called Stiefel-Whitney classes. The next Stiefel-Whitney class measures a more refined sort of orientability called a spin structure, and the higher Stiefel-Whitney classes measure whether the vector bundle looks more and more like a product vector bundle over succesively higher dimensional subspaces of the base space. Cohomological invariants of vector bundles such as these with nice general properties are known as characteristic classes. It turns out that Stiefel-Whitney classes generate all characteristic classes for ordinary cohomology with Z2 coefficients, but with Z coefficients there are others, called Pontryagin and Euler classes, the latter being related to the Euler characteristic. Although characteristic classes do not come close to distinguishing all the different vector bundles over a given base space, except in a few low dimensional cases, they have still proved themselves to be quite useful objects.

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the unit 2 sphere S in R . At each point x∈ S there is a tangent plane Px. This is a 2 dimensional vector space with the point

x as its zero vector 0x. Vectors vx ∈ Px are thought of as arrows with their tail at x . If we regard a vector vx in Px as a vector inR3, then the standard convention in linear algebra would be to identify vx with all its parallel translates, and in particular with the unique translate τ(vx) having its tail at the origin in R3. The association vx

,

τ(vx) defines a

τ

x v

Px x

vx ( )

function τ : T S2

R3 where T S2 is the set of all tangent vectors vx as x ranges over S2. This function τ is surjective but certainly not injective, as every nonzero vector in R3 occurs as τ(vx) for infinitely many x , in fact for all x in a great circle in S2. Moreover τ(0x)= 0 for all x ∈ S2, so τ−1(0) is a whole sphere. On the other hand, the function T S2

S2×R3, vx

,

(x, τ(vx)) , is injective, and can be used to topologize T S2 as a subspace of S2×R3, namely the subspace consisting of pairs (x, v) with v orthogonal to x .

Thus T S2 is first of all a topological space, and secondly it is the disjoint union of all the vector spaces Px for x∈ S2. One can think of T S2 as a continuous family of vector spaces parametrized by points of S2.

The simplest continuous family of 2 dimensional vector spaces parametrized by points of S2 is of course the product S2×R2. Is that what T S2 really is? More precisely we can ask whether there is a homeomorphism h : T S2

S2×R2 that takes

each plane Px to the plane {x}×R2 by a vector space isomorphism. If we had such an h , then for each fixed nonzero vector v∈ R2 the family of vectors vx= h−1(x, v) would be a continuous field of nonzero tangent vectors to S2. It is a classical theorem in algebraic topology that no such vector field exists. (See §2.2 for a proof using

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techniques from this book.) So T S2 is genuinely twisted, and is not just a disguised form of the product S2×R2.

Dropping down a dimension, one could consider in similar fashion the space T S1 of tangent vectors to the unit circle S1 in R2. In this case there is a continuous field vx of nonzero tangent vectors to S1, obtained by regarding

points x ∈ S1 as unit complex numbers and letting vx be the translation of the vector ix that has its tail at x . This leads to a

vx x

homeomorphism S1×R

T S1 taking (x, t) to tvx, with {x}×R going to the tangent line at x by a linear isomorphism. Thus T S1 really is equivalent to the product S1×R1.

Moving up to S3, the unit sphere inR4, the space T S3 of tangent vectors is again equivalent to the product S3×R3. RegardingR4as the quaternions, an equivalence is the homeomorphism S3×R3

T S3 sending (x, (t1, t2, t3)) to the translation of the vector t1ix+ t2jx+ t3kx having its tail at x . A similar construction using Cayley octonions shows that T S7is equivalent to S7×R7. It is a rather deep theorem, proved in§2.3, that S1, S3, and S7 are the only spheres whose tangent bundle is equivalent to a product.

Although T Sn is not usually equivalent to the product Sn×Rn, there is a sense in which this is true locally. Take the case of the 2 sphere for example. For a point x ∈ S2 let P be the translate of the tangent plane Px that passes through the ori- gin. For points y∈ S2 that are sufficiently close to x the map πy: Py

P sending

a tangent vector vy to the orthogonal projection of τ(vy) onto P is a linear iso- morphism. This is true in fact for any y on the same side of P as x . Thus for y in a suitable neighborhood U of x in S2 the map (y, vy)

,

(y, πy(vy)) is a

homeomorphism with domain the subspace of T S2 consisting of tangent vectors at points of U and with range the product U×P . Furthermore this homeomorphism has the key property of restricting to a linear isomorphism from Py onto P for each y ∈ U . A convenient way of rephrasing this situation, having the virtue of easily gen- eralizing, is to let p : T S2

S2 be the map (x, vx)

,

x , and then we have a homeo- morphism p−1(U)

U×P that restricts to a linear isomorphism p−1(y)

{y}×P

for each y∈ U .

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1.1 Basic Definitions and Constructions

Throughout the book we use the word map to mean a continuous function.

An n dimensional vector bundle is a map p : E

B together with a real vector space structure on p−1(b) for each b ∈ B , such that the following local triviality condition is satisfied: There is a cover of B by open sets Uα for each of which there exists a homeomorphism hα: p−1(Uα)

Uα×Rntaking p−1(b) to{b}×Rnby a vec- tor space isomorphism for each b∈ Uα. Such an hα is called a local trivialization of the vector bundle. The space B is called the base space, E is the total space, and the vector spaces p−1(b) are the fibers. Often one abbreviates terminology by just calling the vector bundle E , letting the rest of the data be implicit.

We could equally well takeC in place of R as the scalar field, obtaining the notion of a complex vector bundle. We will focus on real vector bundles in this chapter.

Usually the complex case is entirely analogous. In the next chapter complex vector bundles will play the larger role, however.

Here are some examples of vector bundles:

(1) The product or trivial bundle E = B×Rn with p the projection onto the first factor.

(2) If we let E be the quotient space of I×R under the identifications (0, t) ∼ (1, −t), then the projection I×R

I induces a map p : E

S1which is a 1 dimensional vector bundle, or line bundle. Since E is homeomorphic to a M¨obius band with its boundary circle deleted, we call this bundle the M¨obius bundle.

(3) The tangent bundle of the unit sphere Sn in Rn+1, a vector bundle p : E

Sn

where E = { (x, v) ∈ Sn×Rn+1 | x ⊥ v } and we think of v as a tangent vector to Sn by translating it so that its tail is at the head of x , on Sn. The map p : E

Sn

sends (x, v) to x . To construct local trivializations, choose any point x ∈ Sn and let Ux ⊂ Sn be the open hemisphere containing x and bounded by the hyperplane through the origin orthogonal to x . Define hx: p−1(Ux)

Ux×p−1(x) ≈ Ux×Rn by hx(y, v) = (y, πx(v)) where πx is orthogonal projection onto the hyperplane p−1(x) . Then hx is a local trivialization since πx restricts to an isomorphism of p−1(y) onto p−1(x) for each y∈ Ub.

(4) The normal bundle to Sn in Rn+1, a line bundle p : E

Sn with E consisting of pairs (x, v)∈ Sn×Rn+1 such that v is perpendicular to the tangent plane to Sn at x , or in other words, v = tx for some t ∈ R. The map p : E

Sn is again given by p(x, v)= x . As in the previous example, local trivializations hx: p−1(Ux)

Ux×R

can be obtained by orthogonal projection of the fibers p−1(y) onto p−1(x) for y Ux.

(5) Real projective n space RPn is the space of lines in Rn+1 through the origin.

Since each such line intersects the unit sphere Sn in a pair of antipodal points, we

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can also regard RPn as the quotient space of Sn in which antipodal pairs of points are identified.. The canonical line bundle p : E

RPn has as its total space E the subspace ofRPn×Rn+1 consisting of pairs (`, v) with v∈ `, and p(`, v) = `. Again local trivializations can be defined by orthogonal projection.

There is also an infinite-dimensional projective space RP which is the union of the finite-dimensional projective spaces RPn under the inclusions RPn ⊂ RPn+1 coming from the natural inclusions Rn+1 ⊂ Rn+2. The topology we use on RP is the weak or direct limit topology, for which a set inRP is open iff it intersects each RPn in an open set. The inclusions RPn ⊂ RPn+1 induce corresponding inclusions of canonical line bundles, and the union of all these is a canonical line bundle over RP, again with the direct limit topology. Local trivializations work just as in the finite-dimensional case.

(6) The canonical line bundle over RPn has an orthogonal complement, the space E = { (`, v) ∈ RPn×Rn+1 | v ⊥ ` }. The projection p : E

RPn, p(`, v) = `,

is a vector bundle with fibers the orthogonal subspaces `, of dimension n . Local trivializations can be obtained once more by orthogonal projection.

A natural generalization ofRPnis the so-called Grassmann manifold Gk(Rn) , the space of all k dimenional planes through the origin inRn. The topology on this space will be defined precisely in§1.2, along with a canonical k dimensional vector bundle over it consisting of pairs (`, v) where ` is a point in Gk(Rn) and v is a vector in

` . This too has an orthogonal complement, an (n− k) dimensional vector bundle consisting of pairs (`, v) with v orthogonal to ` .

An isomorphism between vector bundles p1: E1

B and p2: E2

B over the same base space B is a homeomorphism h : E1

E2 taking each fiber p−11 (b) to the cor- responding fiber p2−1(b) by a linear isomorphism. Thus an isomorphism preserves all the structure of a vector bundle, so isomorphic bundles are often regarded as the same. We use the notation E1≈ E2 to indicate that E1 and E2 are isomorphic.

For example, the normal bundle of SninRn+1 is isomorphic to the product bun- dle Sn×R by the map (x, tx)

,

(x, t) . The tangent bundle to S1 is also isomorphic to the trivial bundle S1×R, via (e, ite)

,

(e, t) , for e∈ S1 and t∈ R.

As a further example, the M¨obius bundle in (2) above is isomorphic to the canon- ical line bundle over RP1≈ S1. Namely, RP1 is swept out by a line rotating through an angle of π , so the vectors in these lines sweep out a rectangle [0, π ]×R with the two ends{0}×R and {π}×R identified. The identification is (0, x) ∼ (π, −x) since rotating a vector through an angle of π produces its negative.

One can sometimes distinguish nonisomorphic bundles by looking at the comple- ment of the zero section since any vector bundle isomorphism h : E1

E2 must take

the zero section of E1 onto the zero section of E2, hence the complements of the zero sections in E1 and E2must be homeomorphic. For example, the M¨obius bundle is not

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isomorphic to the product bundle S1×R since the complement of the zero section in the M¨obius bundle is connected while for the product bundle the complement of the zero section is not connected. This method for distinguishing vector bundles can also be used with more refined topological invariants such as homology groups.

Sections

A section of a vector bundle p : E

B is a map s : B

E assigning to each b∈ B a vector s(b) in the fiber p−1(b) . The condition s(b)∈ p−1(b) can also be written as ps=11 , the identity map of B . Every vector bundle has a canonical section, the zero section whose value is the zero vector in each fiber. We often identity the zero section with its image, a subspace of E which projects homeomorphically onto B by p .

One can sometimes distinguish nonisomorphic bundles by looking at the com- plement of the zero section since any vector bundle isomorphism h : E1

E2 must

take the zero section of E1 onto the zero section of E2, so the complements of the zero sections in E1 and E2 must be homeomorphic. For example, we can see that the M¨obius bundle is not isomorphic to the product bundle S1×R since the complement of the zero section is connected for the M¨obius bundle but not for the product bundle.

At the other extreme from the zero section would be a section whose values are all nonzero. Not all vector bundles have such a section. Consider for example the tangent bundle to Sn. Here a section is just a tangent vector field to Sn. As we shall show in§2.2, Sn has a nonvanishing vector field iff n is odd. From this it follows that the tangent bundle of Sn is not isomorphic to the trivial bundle if n is even and nonzero, since the trivial bundle obviously has a nonvanishing section, and an isomorphism between vector bundles takes nonvanishing sections to nonvanishing sections.

In fact, an n dimensional bundle p : E

B is isomorphic to the trivial bundle iff it has n sections s1,··· , sn such that the vectors s1(b),··· , sn(b) are linearly inde- pendent in each fiber p−1(b) . In one direction this is evident since the trivial bundle certainly has such sections and an isomorphism of vector bundles takes linearly inde- pendent sections to linearly independent sections. Conversely, if one has n linearly independent sections si, the map h : B×Rn

E given by h(b, t1,··· , tn)=P

itisi(b) is a linear isomorphism in each fiber, and is continuous since its composition with a local trivialization p−1(U)

U×Rn is continuous. Hence h is an isomorphism by the following useful technical result:

L

emma 1.1. A continuous map h : E1

E2 between vector bundles over the same base space B is an isomorphism if it takes each fiber p−11 (b) to the corresponding fiber p−12 (b) by a linear isomorphism.

P

roof: The hypothesis implies that h is one-to-one and onto. What must be checked is that h−1 is continuous. This is a local question, so we may restrict to an open set

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U ⊂ B over which E1 and E2 are trivial. Composing with local trivializations reduces to the case of an isomorphism h : U×Rn

U×Rn of the form h(x, v)= (x, gx(v)) . Here gx is an element of the group GLn(R) of invertible linear transformations of Rn, and gx depends continuously on x . This means that if gx is regarded as an n×n matrix, its n2 entries depend continuously on x . The inverse matrix gx−1 also depends continuously on x since its entries can be expressed algebraically in terms of the entries of gx, namely, g−1x is 1/(det gx) times the classical adjoint matrix of gx. Therefore h−1(x, v)= (x, g−1x (v)) is continuous. tu

As an example, the tangent bundle to S1 is trivial because it has the section (x1, x2)

,

(−x2, x1) for (x1, x2) ∈ S1. In terms of complex numbers, if we set z= x1+ ix2 then this section is z

,

iz since iz= −x2+ ix1.

There is an analogous construction using quaternions instead of complex num- bers. Quaternions have the form z= x1+ix2+jx3+kx4, and form a division algebra H via the multiplication rules i2 = j2 = k2 = −1, ij = k, jk = i, ki = j , ji = −k, kj= −i, and ik = −j . If we identify H with R4 via the coordinates (x1, x2, x3, x4) , then the unit sphere is S3 and we can define three sections of its tangent bundle by the formulas

z

,

iz or (x1, x2, x3, x4)

,

(−x2, x1,−x4, x3)

z

,

jz or (x1, x2, x3, x4)

,

(−x3, x4, x1,−x2)

z

,

kz or (x1, x2, x3, x4)

,

(−x4,−x3, x2, x1)

It is easy to check that the three vectors in the last column are orthogonal to each other and to (x1, x2, x3, x4) , so we have three linearly independent nonvanishing tangent vector fields on S3, and hence the tangent bundle to S3 is trivial.

The underlying reason why this works is that quaternion multiplication satisfies

|zw| = |z||w|, where |·| is the usual norm of vectors in R4. Thus multiplication by a quaternion in the unit sphere S3 is an isometry ofH. The quaternions 1, i, j, k form the standard orthonormal basis forR4, so when we multiply them by an arbitrary unit quaternion z∈ S3 we get a new orthonormal basis z, iz, jz, kz .

The same constructions work for the Cayley octonions, a division algebra struc- ture on R8. Thinking of R8 as H×H, multiplication of octonions is defined by (z1, z2)(w1, w2)= (z1w1−w2z2, z2w1+w2z1) and satisfies the key property|zw| =

|z||w|. This leads to the construction of seven orthogonal tangent vector fields on the unit sphere S7, so the tangent bundle to S7 is also trivial. As we shall show in

§2.3, the only spheres with trivial tangent bundle are S1, S3, and S7.

Another way of characterizing the trivial bundle E≈ B×Rnis to say that there is a continuous projection map E

Rnwhich is a linear isomorphism on each fiber, since such a projection together with the bundle projection E

B gives an isomorphism E≈ B×Rn, by Lemma 1.1.

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Direct Sums

Given two vector bundles p1: E1

B and p2: E2

B over the same base space B , we would like to create a third vector bundle over B whose fiber over each point of B is the direct sum of the fibers of E1 and E2 over this point. This leads us to define the direct sum of E1 and E2 as the space

E1E2= { (v1, v2)∈ E1×E2| p1(v1)= p2(v2)}

There is then a projection E1⊕E2

B sending (v1, v2) to the point p1(v1)= p2(v2) . The fibers of this projection are the direct sums of the fibers of E1 and E2, as we wanted. For a relatively painless verification of the local triviality condition we make two preliminary observations:

(a) Given a vector bundle p : E

B and a subspace A ⊂ B , then p : p−1(A)

A is

clearly a vector bundle. We call this the restriction of E over A .

(b) Given vector bundles p1: E1

B1 and p2: E2

B2, then p1×p2: E1×E2

B1×B2

is also a vector bundle, with fibers the products p1−1(b1)×p2−1(b2) . For if we have local trivializations hα: p1−1(Uα)

Uα×Rn and hβ: p2−1(Uβ)

Uβ×Rm for E1 and E2, then hα×hβ is a local trivialization for E1×E2.

Then if E1 and E2 both have the same base space B , the restriction of the product E1×E2 over the diagonal B= {(b, b) ∈ B×B} is exactly E1⊕E2.

The direct sum of two trivial bundles is again a trivial bundle, clearly, but the direct sum of nontrivial bundles can also be trivial. For example, the direct sum of the tangent and normal bundles to Sn in Rn+1 is the trivial bundle Sn×Rn+1 since elements of the direct sum are triples (x, v, tx)∈ Sn×Rn+1×Rn+1 with x⊥ v , and the map (x, v, tx)

,

(x, v+tx) gives an isomorphism of the direct sum bundle with Sn×Rn+1. So the tangent bundle to Snis stably trivial: it becomes trivial after taking the direct sum with a trivial bundle.

As another example, the direct sum E⊕E of the canonical line bundle E

RPn

with its orthogonal complement, defined in example (6) above, is isomorphic to the trivial bundle RPn×Rn+1 via the map (`, v, w)

,

(`, v+ w) for v ∈ ` and w ⊥ `.

Specializing to the case n= 1, the bundle E is isomorphic to E itself by the map that rotates each vector in the plane by 90 degrees. We noted earlier that E is isomorphic to the M¨obius bundle over S1 = RP1, so it follows that the direct sum of the M¨obius bundle with itself is the trivial bundle. To see this geometrically, embed the M¨obius bundle in the product bundle S1×R2by taking the line in the fiber{θ}×R2that makes an angle of θ/2 with the x axis, and then the orthogonal lines in the fibers form a second copy of the M¨obius bundle, giving a decomposition of the product S1×R2 as the direct sum of two M¨obius bundles.

E

xample: The tangent bundle of real projective space. Starting with the isomor- phism Sn×Rn+1 ≈ T Sn⊕NSn, where NSn is the normal bundle of Sn in Rn+1,

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suppose we factor out by the identifications (x, v)∼ (−x, −v) on both sides of this isomorphism. Applied to T Sn this identification yields TRPn, the tangent bundle to RPn. This is saying that a tangent vector toRPn is equivalent to a pair of antipodal tangent vectors to Sn. A moment’s reflection shows this to be entirely reasonable, although a formal proof would require a significant digression on what precisely tan- gent vectors to a smooth manifold are, a digression we shall skip here. What we will show is that even though the direct sum of TRPn with a trivial line bundle may not be trivial as it is for a sphere, it does split in an interesting way as a direct sum of nontrivial line bundles.

In the normal bundle NSn the identification (x, v)∼ (−x, −v) can be written as (x, tx)∼ (−x, t(−x)). This identification yields the product bundle RPn×R since the section x

,

(−x, −x) is well-defined in the quotient. Now let us consider the identification (x, v) ∼ (−x, −v) in Sn×Rn+1. This identification respects the co- ordinate factors of Rn+1, so the quotient is the direct sum of n+ 1 copies of the line bundle E over RPn obtained by making the identifications (x, t) ∼ (−x, −t) in Sn×R. The claim is that E is just the canonical line bundle over RPn. To see this, let us identify Sn×R with NSn by the isomorphism (x, t)

,

(x, tx) . hence (−x, −t)

,

((−x, (−t)(−x)) = (−x, tx). Thus we have the identification (x, tx) ∼ (−x, tx) in NSn. The quotient is the canonical line bundle overRPn since the iden- tifications x∼ −x in the first coordinate give lines through the origin in Rn+1, and in the second coordinate there are no identifications so we have well-defined vectors tx in these lines.

Thus we have shown that the tangent bundle TRPn is stably isomorphic to the direct sum of n+ 1 copies of the canonical line bundle over RPn. When n= 3, for example, TRP3 is trivial since the three linearly independent tangent vector fields on S3 defined earlier in terms of quaternions pass down to linearly independent tangent vector fields on the quotientRP3. Hence the direct sum of four copies of the canonical line bundle over RP3 is trivial. Similarly using octonions we can see that the direct sum of eight copies of the canonical line bundle over RP7 is trivial. In§2.5 we will determine when the sum of k copies of the canonical line bundle over RPnis at least stably trivial. The answer turns out to be rather subtle: This happens exactly when k is a multiple of 2ϕ(n) where ϕ(n) is the number of integers i in the range 0 < i≤ n with i congruent to 0 , 1 , 2 , or 4 modulo 8 . For n = 3, 7 this gives 2ϕ(n) = 4, 8, the same numbers we obtained from the triviality of TRP3 and TRP7. If there were a 16 dimensional division algebra after the octonions then one might expect the sum of 16 copies of the canonical line bundle over RP15 to be trivial. However this is not the case since 2ϕ(15)= 27= 128.

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Inner Products

An inner product on a vector bundle p : E

B is a map h , i : EE

R which

restricts in each fiber to an inner product, a positive definite symmetric bilinear form.

P

roposition 1.2. An inner product exists for a vector bundle p : E

B if B is compact Hausdorff or more generally paracompact.

The definition of paracompactness we are using is that a space X is paracom- pact if it is Hausdorff and every open cover has a subordinate partition of unity, a collection of maps ϕβ: X

[0, 1] each supported in some set of the open cover, and with P

βϕβ = 1, only finitely many of the ϕβ being nonzero near each point of X . Constructing such functions is easy when X is compact Hausdorff, using Urysohn’s Lemma. This is done in the appendix to this chapter, where we also show that certain classes of noncompact spaces are paracompact. Most spaces that arise naturally in algebraic topology are paracompact.

P

roof: An inner product for p : E

B can be constructed by first using local trivial- izations hα: p−1(Uα)

Uα×Rn, to pull back the standard inner product in Rn to an inner producth·, ·iα on p−1(Uα) , then setting hv, wi =P

βϕβp(v)hv, wiα(β) where β} is a partition of unity with the support of ϕβ contained in Uα(β). tu

In the case of complex vector bundles one can construct Hermitian inner products in the same way.

In linear algebra one can show that a vector subspace is always a direct summand by taking its orthogonal complement. We will show now that the corresponding result holds for vector bundles over a paracompact base. A vector subbundle of a vector bundle p : E

B has the natural definition: a subspace E0⊂ E intersecting each fiber of E in a vector subspace, such that the restriction p : E0

B is a vector bundle.

P

roposition 1.3. If E

B is a vector bundle over a paracompact base B and E0⊂ E is a vector subbundle, then there is a vector subbundle E0⊂ E such that E0E0 ≈ E .

P

roof: With respect to a chosen inner product on E , let E0 be the subspace of E which in each fiber consists of all vectors orthogonal to vectors in E0. We claim that the natural projection E0

B is a vector bundle. If this is so, then E0⊕E0 is isomorphic to E via the map (v, w)

,

v+ w , using Lemma 1.1.

To see that E0 satisfies the local triviality condition for a vector bundle, note first that we may assume E is the product B×Rn since the question is local in B . Since E0 is a vector bundle, of dimension m say, it has m independent local sections b

,

(b, si(b)) near each point b0 ∈ B . We may enlarge this set of m independent local sections of E0 to a set of n independent local sections b

,

(b, si(b)) of E

by choosing sm+1,··· , sn first in the fiber p−1(b0) , then taking the same vectors for all nearby fibers, since if s1,··· , sm, sm+1,··· , sn are independent at b0, they will

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remain independent for nearby b by continuity of the determinant function. Apply the Gram-Schmidt orthogonalization process to s1,··· , sm, sm+1,··· , sn in each fiber, using the given inner product, to obtain new sections si0. The explicit formulas for the Gram-Schmidt process show the si0’s are continuous, and the first m of them are a basis for E0 in each fiber. The sections si0 allow us to define a local trivialization h : p−1(U)

U×Rn with h(b, si0(b)) equal to the ith standard basis vector of Rn. This h carries E0 to U×Rm and E0 to U×Rn−m, so h ||E0 is a local trivialization of

E0. tu

Note that when the subbundle E0 is equal to E itself, the last part of the proof shows that for any vector bundle with an inner product it is always possible to choose local trivializations that carry the inner product to the standard inner product, so the local trivializations are by isometries.

We have seen several cases where the sum of two bundles, one or both of which may be nontrivial, is the trivial bundle. Here is a general result result along these lines:

P

roposition 1.4. For each vector bundle E

B with B compact Hausdorff there exists a vector bundle E0

B such that E⊕E0 is the trivial bundle.

This can fail when B is noncompact. An example is the canonical line bundle over RP, as we shall see in Example 3.6.

P

roof: To motivate the construction, suppose first that the result holds and hence that E is a subbundle of a trivial bundle B×RN. Composing the inclusion of E into this product with the projection of the product onto RN yields a map E

RN that

is a linear injection on each fiber. Our strategy will be to reverse the logic here, first constructing a map E

RN that is a linear injection on each fiber, then showing that this gives an embedding of E in B×RN as a direct summand.

Each point x ∈ B has a neighborhood Ux over which E is trivial. By Urysohn’s Lemma there is a map ϕx: B

[0, 1] that is 0 outside Ux and nonzero at x . Letting x vary, the sets ϕ−1x (0, 1] form an open cover of B . By compactness this has a finite subcover. Let the corresponding Ux’s and ϕx’s be relabeled Ui and ϕi. Define gi: E

Rn by gi(v)= ϕi(p(v))[πihi(v)] where p is the projection E

B and πihi

is the composition of a local trivialization hi: p−1(Ui)

Ui×Rn with the projection πi to Rn. Then gi is a linear injection on each fiber over ϕ−1i (0, 1] , so if we make the various gi’s the coordinates of a map g : E

RN with RN a product of copies of Rn, then g is a linear injection on each fiber.

The map g is the second coordinate of a map f : E

B×RN with first coordinate p . The image of f is a subbundle of the product B×RN since projection of RN onto the ith Rn factor gives the second coordinate of a local trivialization over ϕ−1i (0, 1] . Thus we have E isomorphic to a subbundle of B×RN so by preceding proposition there is a complementary subbundle E0 with E⊕E0 isomorphic to B×RN. tu

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Tensor Products

In addition to direct sum, a number of other algebraic constructions with vec- tor spaces can be extended to vector bundles. One which is particularly important for K–theory is tensor product. For vector bundles p1: E1

B and p2: E2

B , let

E1⊗E2, as a set, be the disjoint union of the vector spaces p−11 (x)⊗p−12 (x) for x∈ B . The topology on this set is defined in the following way. Choose isomorphisms hi: pi−1(U)

U×Rni for each open set U⊂ B over which E1 and E2 are trivial. Then a topology TU on the set p−11 (U)⊗p−12 (U) is defined by letting the fiberwise tensor product map h1h2: p−11 (U)⊗p2−1(U)

U×(Rn1⊗Rn2) be a homeomorphism. The topology TU is independent of the choice of the hi’s since any other choices are ob- tained by composing with isomorphisms of U×Rni of the form (x, v)

,

(x, gi(x)(v))

for continuous maps gi: U

GLni(R), hence h1h2 changes by composing with analogous isomorphisms of U×(Rn1⊗Rn2) whose second coordinates g1g2 are continuous maps U

GLn1n2(R), since the entries of the matrices g1(x)g2(x) are the products of the entries of g1(x) and g2(x) . When we replace U by an open sub- set V , the topology on p−11 (V )⊗p−12 (V ) induced by TU is the same as the topology TV since local trivializations over U restrict to local trivializations over V . Hence we get a well-defined topology on E1⊗E2 making it a vector bundle over B .

There is another way to look at this construction that takes as its point of depar- ture a general method for constructing vector bundles we have not mentioned previ- ously. If we are given a vector bundle p : E

B and an open cover{Uα} of B with lo- cal trivializations hα: p−1(Uα)

Uα×Rn, then we can reconstruct E as the quotient space of the disjoint union `

α(Uα×Rn) obtained by identifying (x, v) ∈ Uα×Rn with hβh−1α (x, v) ∈ Uβ×Rn whenever x ∈ Uα∩ Uβ. The functions hβh−1α can be viewed as maps gβα: Uα∩ Uβ

GLn(R). These satisfy the ‘cocycle condition’

gγβgβα = gγα on Uα∩ Uβ∩ Uγ. Any collection of ‘gluing functions’ gβα satisfying this condition can be used to construct a vector bundle E

B .

In the case of tensor products, suppose we have two vector bundles E1

B and

E2

B . We can choose an open cover{Uα} with both E1 and E2 trivial over each Uα, and so obtain gluing functions gβαi : Uα∩ Uβ

GLni(R) for each Ei. Then the gluing functions for the bundle E1⊗E2 are the tensor product functions gβα1 gβα2 assigning to each x∈ Uα∩ Uβ the tensor product of the two matrices g1βα(x) and g2βα(x) .

It is routine to verify that the tensor product operation for vector bundles over a fixed base space is commutative, associative, and has an identity element, the trivial line bundle. It is also distributive with respect to direct sum.

If we restrict attention to line bundles, then the set Vect1(B) of isomorphism classes of one-dimensional vector bundles over B is an abelian group with respect to the tensor product operation. The inverse of a line bundle E

B is obtained by replacing its gluing matrices gβα(x)∈ GL1(R) with their inverses. The cocycle con- dition is preserved since 1×1 matrices commute. If we give E an inner product, we

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