"Version 1.3, July 2001
Allen Hatcher
Copyright c 2001 by Allen Hatcher
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Table of Contents
Chapter 1. Vector Bundles
1.1. Basic Definitions and Constructions
. . . . . . . . . . . . 1 Sections 3. Direct Sums 5. Pullback Bundles 5. Inner Products 7. Subbundles 8. Tensor Products 9. Associated Bundles 11.
1.2. Classifying Vector Bundles
. . . . . . . . . . . . . . . . .
12
The Universal Bundle 12. Vector Bundles over Spheres 16. Orientable Vector Bundles 21. A Cell Structure on Grassmann Manifolds 22. Appendix: Paracompactness 24.
Chapter 2. Complex K-Theory
2.1. The Functor K(X) 2.2. Bott Periodicity
. . . . . . . . . . . . . . . . . . . . . . . 28 Ring Structure 31. Cohomological Properties 32. . . . . . . . . . . . . . . . . . . . . . . . . 38
Clutching Functions 38. Linear Clutching Functions 43. Conclusion of the Proof 45.
2.3. Adams’ Hopf Invariant One Theorem
Adams Operations 51. The Splitting Principle 55.
. . . . . . . . . . .
48
2.4. Further Calculations
The Thom Isomorphism 61.
. . . . . . . . . . . . . . . . . . . . .
61
Chapter 3. Characteristic Classes
3.1. Stiefel-Whitney and Chern Classes
Applications of w1 and c1 72. . . . . . . . . . . . . 63 Axioms and Construction 64. Cohomology of Grassmannians 69.
3.2. The Chern Character
The J–Homomorphism 76.
. . . . . . . . . . . . . . . . . . . . .
73
3.3. Euler and Pontryagin Classes
. . . . . . . . . . . . . . . .
83
The Euler Class 87. Pontryagin Classes 90.
1. Basic Definitions and Constructions
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 × Rn taking p −1 (b) to {b}× Rn by a vector space isomorphism for each b ∈ Uα . Such an hα is 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 take C in place of R as the scalar field here, obtaining the notion of a complex vector bundle. If we modify the definition by dropping all references to vector spaces and replace R
n
ture. Here is the basic definition. An n dimensional vector bundle is a map p : E →B
Vector bundles are special sorts of fiber bundles with additional algebraic struc-
called a local trivialization of the vector bundle. The space B is called the base space,
exists a homeomorphism hα : p −1 (Uα )→Uα × F taking p −1 (b) to {b}× F for each b ∈ Uα . Here are some examples of vector bundles:
p : E →B such that there is a cover of B by open sets Uα for each of which there
by an arbitrary space F , then we have the definition of a fiber bundle: a map
(1) The product or trivial bundle E = B × Rn with p the projection onto the first factor. then the projection I × R→I induces a map p : E →S 1 which is a 1 dimensional vector bundle, or line bundle. Since E is homeomorphic to a M¨bius band with its boundary o circle deleted, we call this bundle the M¨bius bundle. o (3) The tangent bundle of the unit sphere S n in Rn+1 , a vector bundle p : E →S n S n by translating it so that its tail is at the head of x , on S n . The map p : E →S n where E = { (x, v) ∈ S n × Rn+1 | x ⊥ v } and we think of v as a tangent vector to (2) If we let E be the quotient space of I × R under the identifications (0, t) ∼ (1, −t) ,
2
Chapter 1
Vector Bundles
sends (x, v) to x . To construct local trivializations, choose any point b ∈ S n and let Ub ⊂ S n be the open hemisphere containing b and bounded by the hyperplane through the origin orthogonal to b . Define hb : p −1 (Ub )→Ub × p −1 (b) ≈ Ub × Rn by hb (x, v) = (x, πb (v)) where πb is orthogonal projection onto the tangent plane p −1 (x) onto p −1 (b) for each x ∈ Ub . p −1 (b) . Then hb is a local trivialization since πb restricts to an isomorphism of
pairs (x, v) ∈ S n × Rn+1 such that v is perpendicular to the tangent plane to S n at
(4) The normal bundle to S n in Rn+1 , a line bundle p : E →S n with E consisting of
in the previous example, local trivializations hb : p −1 (Ub )→Ub × R can be obtained by orthogonal projection of the fibers p −1 (x) onto p −1 (b) for x ∈ Ub . (5) The canonical line bundle p : E →RPn . Thinking of RPn as the space of lines in Rn+1 through the origin, E is the subspace of RPn × Rn+1 consisting of pairs ( , v) with v ∈ , and p( , v) = . Again local trivializations can be defined by orthogonal projection. We could also take n = ∞ and get the canonical line bundle E →RP∞ . line bundle. The projection p : E ⊥ →RPn , p( , v) =
⊥
x , i.e., v = tx for some t ∈ R . The map p : E →S n is again given by p(x, v) = x . As
(6) The orthogonal complement E ⊥ = { ( , v) ∈ RPn × Rn+1 | v ⊥ the orthogonal subspaces
} of the canonical
, is a vector bundle with fibers
, of dimension n . Local trivializations can be obtained
once more by orthogonal projection.
−1 base space B is a homeomorphism h : E1 →E2 taking each fiber p1 (b) to the cor-
An isomorphism between vector bundles p1 : E1 →B and p2 : E2 →B over the same
−1 responding fiber p2 (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 S n in Rn+1 is isomorphic to the product bundle S n × R by the map (x, tx) (x, t) . The tangent bundle to S 1 is also isomorphic (eiθ , t) , for eiθ ∈ S 1 and t ∈ R . to the trivial bundle S 1 × R , via (eiθ , iteiθ )
As a further example, the M¨bius bundle in (2) above is isomorphic to the canono ical line bundle over RP1 ≈ S 1 . 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. The zero section of a vector bundle p : E →B is the union of the zero vectors in all the fibers. This is a subspace of E which projects homeomorphically onto B by p . Moreover, E deformation retracts onto its zero section via the homotopy ft (v) = (1 − t)v given by scalar multiplication of vectors v ∈ E . Thus all vector bundles over B have the same homotopy type. ment of the zero section since any vector bundle isomorphism h : E1 →E2 must take One can sometimes distinguish nonisomorphic bundles by looking at the comple-
Basic Definitions and Constructions
Section 1.1
3
the zero section of E1 onto the zero section of E2 , hence the complements of the zero o sections in E1 and E2 must be homeomorphic. For example, the M¨bius bundle is not isomorphic to the product bundle S 1 × R since the complement of the zero section in the M¨bius bundle is connected while for the product bundle the complement of o the zero section is not connected. This method for distinguishing vector bundles can also be used with more refined topological invariants such as Hn in place of H0 . We shall denote the set of isomorphism classes of n dimensional real vector bundles over B by Vectn (B) , and its complex analogue by Vectn (B) . For those who C worry about set theory, we are using the term ‘set’ here in a naive sense. It follows from Theorem 1.8 later in the chapter that Vectn (B) and Vectn (B) are indeed sets in C the strict sense when B is paracompact. o For example, Vect1 (S 1 ) contains exactly two elements, the M¨bius bundle and the product bundle. This will be a rather trivial application of later theory, but it might be an interesting exercise to prove it now directly from the definitions.
Sections
s(b) ∈ p −1 (b) for all b ∈ B . We have already mentioned the zero section, which is the section whose values are all zero. At the other extreme would be a section whose values are all nonzero. Not all vector bundles have such a nonvanishing section. Consider for example the tangent bundle to S n . Here a section is just a tangent vector field to S n . One of the standard first applications of homology theory is the theorem that S n has a nonvanishing vector field iff n is odd. From this it follows that the tangent bundle of S n 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 each fiber p −1 (b) . For if one has such sections si , the map h : B × Rn →E given by h(b, t1 , ··· , tn ) =
i ti si (b)
A section of a bundle p : E →B is a map s : B →E such that ps = 1 , or equivalently, 1
it has n sections s1 , ··· , sn such that s1 (b), ··· , sn (b) are linearly independent in is a linear isomorphism in each fiber, and is continuous,
as can be verified by composing with a local trivialization p −1 (U )→U × Rn . Hence h A continuous map h : E1 →E2 between vector bundles over the same
is an isomorphism by the following useful technical result:
−1 base space B is an isomorphism if it takes each fiber p1 (b) to the corresponding −1 fiber p2 (b) by a linear isomorphism.
Lemma 1.1.
Proof:
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 to the case of an isomorphism h : U × Rn →U × Rn of the form h(x, v) = (x, gx (v)) . U ⊂ B over which E1 and E2 are trivial. Composing with local trivializations reduces
4
Chapter 1
Vector Bundles
Here gx is an element of the group GLn (R) of invertible linear transformations of Rn which depends continuously on x . This means that if gx is regarded as an n× n
−1 matrix, its n2 entries depend continuously on x . The inverse matrix gx also depends
continuously on x since its entries can be expressed algebraically in terms of the
−1 Therefore h−1 (x, v) = (x, gx (v)) is continuous. −1 entries of gx , namely, gx is 1/(det gx ) times the classical adjoint matrix of gx .
As an example, the tangent bundle to S 1 is trivial because it has the section (x1 , x2 ) (−x2 , x1 ) for (x1 , x2 ) ∈ S 1 . In terms of complex numbers, if we set iz since iz = −x2 + ix1 . z = x1 + ix2 then this section is z
There is an analogous construction using quaternions instead of complex numbers. Quaternions have the form z = x1 + ix2 + jx3 + kx4 , and form a division algebra H via the multiplication rules i2 = j 2 = 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 S 3 and we can define three sections of its tangent bundle by the formulas z z z iz jz kz or or or (x1 , x2 , x3 , x4 ) (x1 , x2 , x3 , x4 ) (x1 , x2 , x3 , x4 ) (−x2 , x1 , −x4 , x3 ) (−x3 , x4 , x1 , −x2 ) (−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 S 3 , and hence the tangent bundle to S 3 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 S 3 is an isometry of H . The quaternions 1, i, j, k form the standard orthonormal basis for R4 , so when we multiply them by an arbitrary unit quaternion z ∈ S 3 we get a new orthonormal basis z, iz, jz, kz . The same constructions work for the Cayley octonions, a division algebra structure on R8 . Thinking of R8 as H× H , multiplication of octonions is defined by (z1 , z2 )(w1 , w2 ) = (z1 w1 − w 2 z2 , z2 w 1 + w2 z1 ) and satisfies the key property |zw| = |z||w| . This leads to the construction of seven orthogonal tangent vector fields on the unit sphere S 7 , so the tangent bundle to S 7 is also trivial. As we shall show in §2.3, the only spheres with trivial tangent bundle are S 1 , S 3 , and S 7 . One final general remark before continuing with our next topic: Another way of characterizing the trivial bundle E ≈ B × Rn is to say that there is a continuous projection map E →Rn which is a linear isomorphism on each fiber, since such a projection together with the bundle projection E →B gives an isomorphism E ≈ B × Rn .
Basic Definitions and Constructions
Section 1.1
5
Direct Sums
As a preliminary to defining a direct sum operation on vector bundles, we make two simple 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
−1 −1 is also a vector bundle, with fibers the products p1 (b1 )× p2 (b2 ) . For if we have
−1 −1 local trivializations hα : p1 (Uα )→Uα × Rn and hβ : p2 (Uβ )→Uβ × Rm for E1 and
E2 , then hα × hβ is a local trivialization for E1 × E2 . Now suppose we are given two vector bundles p1 : E1 →B and p2 : E2 →B over the same base space B . The restriction of the product E1 × E2 over the diagonal B = {(b, b) ∈ B × B} is then a vector bundle, called the direct sum E1 ⊕ E2 →B . Thus E1 ⊕ E2 = { (v1 , v2 ) ∈ E1 × E2 | p1 (v1 ) = p2 (v2 ) } The fiber of E1 ⊕ E2 over a point b ∈ B is the product, or direct sum, of the vector
−1 −1 spaces p1 (b) and p2 (b) .
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 S n in Rn+1 is the trivial bundle S n × Rn+1 since elements of the direct sum are triples (x, v, tx) ∈ S n × Rn+1 × Rn+1 with x ⊥ v , and the map (x, v, tx) S ×R
n n+1
(x, v + tx) gives an isomorphism of the direct sum bundle with
. So the tangent bundle to S n is 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) Specializing to the case n = 1 , both E and E
1 1 ⊥
( , v + w) for v ∈
and w ⊥
.
are isomorphic to the M¨bius bundle o
o over RP = S , so the direct sum of the M¨bius bundle with itself is the trivial bundle. This is just saying that if one takes a slab I × R2 and glues the two faces {0}× R2 and {1}× R2 to each other via a 180 degree rotation of R2 , the resulting vector bundle over S 1 is the same as if the gluing were by the identity map. In effect, one can gradually decrease the angle of rotation of the gluing map from 180 degrees to 0 without changing the vector bundle.
Pullback Bundles
bundles over B into vector bundles over A . Given a vector bundle p : E →B , let Next we describe a procedure for using a map f : A→B to transform vector
6
Chapter 1
Vector Bundles
f ∗ (E) = { (a, v) ∈ A× E | f (a) = p(v) } . This subspace of A× E fits into the communot hard to see that π : f (E)→A is also a vector bundle with fibers
∗
tative diagram at the right where π (a, v) = a and f (a, v) = v . It is
∼ f ∗ f ( E ) −− E −→ − −
π
− − →
p
− − →
of the same dimension as in E . For example, we could say that 1× f ∗ (E) is the restriction of the vector bundle 1 p : A× E →A× B (a, f (a))
A −− B −− −→
f
over the graph of f , {(a, f (a)) ∈ A× B} , which we identify with A via the projection a . The vector bundle f ∗ (E) is called the pullback or induced bundle. v , since the condition As a trivial example, if f is the inclusion of a subspace A ⊂ B , then f ∗ (E) is isomorphic to the restriction p −1 (A) via the map (a, v) f (a) = p(v) just says that v ∈ p case of pullback. pullback of the M¨bius bundle E →S 1 by the two-to-one covering map f : S 1 →S 1 , o An interesting example which is small enough to be visualized completely is the
−1
(a) . So restriction over subspaces is a special
f (z) = z2 . In this case the pullback f ∗ (E) is a two-sheeted covering space of E
which can be thought of as a coat of paint applied to ‘both sides’ of the M¨bius bundle. o Since E has one half-twist, f ∗ (E) has two half-twists, hence is the trivial bundle. More o generally, if En is the pullback of the M¨bius bundle by the map z the trivial bundle for n even and the M¨bius bundle for n odd. o Some elementary properties of pullbacks, whose proofs are one-minute exercises in definition-chasing, are: (i) (f g)∗ (E) ≈ g ∗ (f ∗ (E)) . (iii) f ∗ (E1 ⊕ E2 ) ≈ f ∗ (E1 ) ⊕ f ∗ (E2 ) . (ii) If E1 ≈ E2 then f ∗ (E1 ) ≈ f ∗ (E2 ) . zn , then En is
Now we come to our first important result:
∗ ∗ then the induced bundles f0 (E) and f1 (E) are isomorphic if A is paracompact.
Theorem 1.2.
Given a vector bundle p : E →B and homotopic maps f0 , f1 : A→B ,
All the spaces one ordinarily encounters in algebraic and geometric topology are paracompact, for example compact Hausdorff spaces and CW complexes; see the Appendix to this chapter for more information about this. Let F : A× I →B be a homotopy from f0 to f1 . The restrictions of F ∗ (E) over 1.3. The restrictions of a vector bundle E →X × I over X × {0} and
Proof:
∗ ∗ A× {0} and A× {1} are then f0 (E) and f1 (E) . So the theorem will follow from:
Proposition
Proof:
X × {1} are isomorphic if X is paracompact. We need two preliminary facts: (1) A vector bundle p : E →X × [a, b] is trivial if its restrictions over X × [a, c] and X × [c, b] are both trivial for some c ∈ (a, b) . To see this, let these restrictions be E1 = p −1 (X × [a, c]) and E2 = p −1 (X × [c, b]) , and let h1 : E1 →X × [a, c]× Rn
Basic Definitions and Constructions
Section 1.1
7
the isomorphism X × [c, b]× Rn →X × [c, b]× Rn which on each slice X × {x}× Rn is define a trivialization of E .
p −1 (X × {c}) , but they can be made to agree by replacing h2 by its composition with
and h2 : E2 →X × [c, b]× Rn be isomorphisms. These isomorphisms may not agree on
given by h1 h−1 : X × {c}× Rn →X × {c}× Rn . Once h1 and h2 agree on E1 ∩ E2 , they 2
(2) For a vector bundle p : E →X × I , there exists an open cover {Uα } of X so that each open neighborhoods Ux,1 , ··· , Ux,k in X and a partition 0 = t0 < t1 < ··· < tk = 1 of
restriction p −1 (Uα × I)→Uα × I is trivial. This is because for each x ∈ X we can find [0, 1] such that the bundle is trivial over Ux,i × [ti−1 , ti ] , using compactness of [0, 1] . Then by (1) the bundle is trivial over Uα × I where Uα = Ux,1 ∩ ··· ∩ Ux,k . Now we prove the proposition. By (2), we can choose an open cover {Uα } of X so that E is trivial over each Uα × I . Lemma 1.19 in the Appendix to this chapter asserts that there is a countable cover {Vk }k≥1 of X and a partition of unity {ϕk } with ϕk supported in Vk , such that each Vk is a disjoint union of open sets each contained in some Uα . This means that E is trivial over each Vk × I . so Xk = { (x, ψk (x)) ∈ X × I } , and let pk : Ek →Xk be the restriction of the bunFor k ≥ 0 , let ψk = ϕ1 + ··· + ϕk , wi..."
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