2.3. Adams’ Hopf Invariant One Theorem

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With the hard work of proving Bott Periodicity now behind us, the goal of this

section is to prove Adams’ theorem on the Hopf invariant, with its famous applications

including the nonexistence of division algebras beyond the Cayley octonions:

Theorem 2.14. The following statements are true only for n . 1, 2, 4, and 8:

(a) Rn is a division algebra.

(b) Sn−1 is parallelizable, i.e., there exist n − 1 tangent vector fields to Sn−1 which

are linearly independent at each point, or in other words, the tangent bundle to

Sn−1 is trivial.

(c) Sn−1 is an H–space.

To say that Sn−1 is an H–space means there is a continuous multiplication map

Sn−1_Sn−1!Sn−1 having a two-sided identity element e 2 Sn−1 . This is weaker than

being a topological group since associativity and inverses are not assumed. For example,

S1 , S3 , and S7 are H–spaces by restricting the multiplication of complex numbers,

quaternions, and Cayley octonions to the respective unit spheres, but only S1 and S3

are topological groups since the multiplication of octonions is nonassociative.

A division algebra structure on Rn is a multiplication map Rn_Rn!Rn such

that the maps x,ax and x,xa are linear for each a 2 Rn and invertible if a . 0.

Since we are dealing with linear maps Rn!Rn , invertibility is equivalent to having

trivial kernel, which translates into the statement that the multiplication has no zero

divisors. An identity element is not assumed, but the multiplication can be modified

to produce an identity in the following way. Choose a unit vector e 2 Rn . After

composing the multiplication with an invertible linear map Rn!Rn taking e2 to e

we may assume that e2 . e. Let _ be the map x,xe and _ the map x,ex. The new

product .x;y.,_−1.x._−1.y. then sends .x; e. to _−1.x._−1.e. . _−1.x.e . x,

and similarly it sends .e;y. to y . Since the maps x,ax and x,xa are surjective

for each a . 0, the equations ax . e and xa . e are solvable, so nonzero elements

of the division algebra have multiplicative inverses on the left and right.

The first step in the proof of the theorem is to reduce (a) and (b) to (c):

Adams’ Hopf Invariant One Theorem Section 2.3 49

Lemma 2.15. If Rn is a division algebra, or if Sn−1 is parallelizable, then Sn−1 is

an H–space.

Proof: Having a division algebra structure on Rn with two-sided identity, an H–space

structure on Sn−1 is given by .x;y.,xy=jxyj , which is well-defined since a division

algebra has no zero divisors.

Now suppose that Sn−1 is parallelizable, with tangent vector fields v1; ___ ; vn−1

which are linearly independent at each point of Sn−1 . By the Gram-Schmidt process we

may make the vectors x;v1.x.; ___ ; vn−1.x. orthonormal for all x 2 Sn−1 . We may

assume also that at the first standard basis vector e1 , the vectors v1.e1.; ___ ; vn−1.e1.

are the standard basis vectors e2; ___ ; en, by changing the sign of vn−1 if necessary to

get orientations right, then deforming the vector fields near e1 . Let _x

2 SO.n. send

the standard basis to x;v1.x.; ___ ; vn−1.x.. Then the map .x;y.,_x.y. defines

an H–space structure on Sn−1 with identity element the vector e1 since _e1 is the

identity map and _x.e1. . x for all x. tu

Before proceeding further let us list a few easy consequences of Bott periodicity

which will be needed.

(1) We have already seen that e K.Sn. is Z for n even and 0 for n odd. This comes

from repeated application of the periodicity isomorphism e K.X. _ e K.S2X., _,

__.H −1., the external product with the generator H −1 of eK.S2., where H is

the canonical line bundle over S2 . CP1 . In particular we see that a generator of

e K.S2k. is the k fold external product .H − 1. _ ___ _ .H − 1.. We note also that

the multiplication in e K.S2k. is trivial since this ring is the k fold tensor product

of the ring e K.S2., which has trivial multiplication by Example 2.3. Alternatively,

we can appeal to Example 2.6.

(2) The external product e K.S2k. e K.X.!e K.S2k ^ X. is an isomorphism since it is

an iterate of the periodicity isomorphism.

(3) The external product K.S2k.K.X.!K.S2k_X. is an isomorphism. This follows

from (2) by the same reasoning which showed the equivalence of the reduced

and unreduced forms of Bott periodicity. Since external product is a ring homomorphism,

the isomorphism e K.S2k ^X. _ e K.S2k. e K.X. is a ring isomorphism.

For example, since K.S2k. can be described as the quotient ring Z._.=._2., we

can deduce that K.S2k_S2`. is Z._; _.=._2; _2. where _ and _ are the pullbacks

of generators of e K.S2k. and e K.S2`. under the projections of S2k_S2` onto its

two factors. An additive basis for K.S2k_S2`. is thus f1;_; _;__g.

We can apply the last calculation to show that S2k is not an H–space if k > 0.

Suppose _ : S2k_S2k!S2k is an H–space multiplication. The induced homomorphism

of K–rings then has the form __ : Z..=.2.!Z._; _.=._2; _2.. We claim that

__.. . _._.m__ for some integer m. The composition S2k --!-i S2k_S2k --!-_ S2k

is the identity, where i is the inclusion onto either of the subspaces S2k_feg or

50 Chapter 2 Complex K–Theory

feg_S2k , with e the identity element of the H–space structure. The map i_ for i the

inclusion onto the first factor sends _ to  and _ to 0, so the coefficient of _ in __..

must be 1. Similarly the coefficient of _ must be 1, proving the claim. However, this

leads to a contradiction since it implies that __.2. . ._ . _ .m__.2 . 2__ . 0,

which is impossible since 2 . 0.

There remains the much more difficult problem of showing that Sn−1 is not an

H–space when n is even and different from 2, 4, and 8. The first step is a simple

construction which associates to a map g : Sn−1_Sn−1!Sn−1 a map b g : S2n−1!Sn .

To define this, we regard S2n−1 as @.Dn_Dn. . @Dn_Dn [ Dn_@Dn , and Sn we

take as the union of two disks Dn

. and Dn

− with their boundaries identified. Then

bg is defined on @Dn_Dn by b g.x;y. . jyjg.x;y=jyj. 2 Dn

. and on Dn_@Dn by

b g.x;y. . jxjg.x=jxj;y. 2 Dn

− . Note that b g is well-defined and continuous, even

when jxj or jyj is zero, and b g agrees with g on Sn−1_Sn−1 .

Now we specialize to the case that n is even, or in other words, we replace n

by 2n. For a map f : S4n−1!S2n , let Cf be S2n with a cell e4n attached by f . The

quotient Cf =S2n is then S4n , and since eK

1.S4n. . eK

1.S2n. . 0, the exact sequence

of the pair .Cf ; S2n. becomes a short exact sequence

0 -!e K.S4n. -!e K.Cf . -!e K.S2n. -!0

Let _ 2 e K.Cf . be the image of the generator .H − 1. _ ____.H − 1. of e K.S4n. and

let _ 2 e K.Cf . map to the generator .H − 1. _ ____.H − 1. of e K.S2n.. The element

_2 maps to 0 in e K.S2n. since the square of any element of e K.S2n. is zero. So by

exactness we have _2 . h_ for some integer h. The mod 2 value of h depends only

on f , not on the choice of _, since _ is unique up to adding an integer multiple of

_, and ._ . m_.2 . _2 . 2m__ since _2 . 0. The value of h mod 2 is called the

mod 2 Hopf invariant of f . In fact __ . 0 so h is well-defined in Z not just Z2, as

we will see in x3.2, but for our present purposes the mod 2 value of h suffices.

Lemma 2.16. If g : S2n−1_S2n−1!S2n−1 is an H–space multiplication, then the associated

map bg : S4n−1!S2n has Hopf invariant _1.

Proof: Let e 2 S2n−1 be the identity element for the H–space multiplication, and let

f . b g. In view of the definition of f it is natural to view the characteristic map Ø of

the 4n cell of Cf as a map .D2n_D2n; @.D2n_D2n..!.Cf ; S2n.. In the following

commutative diagram the horizontal maps are the product maps. The diagonal map is

external product, equivalent to the external product e K.S2n. e K.S2n.!e K.S4n., which

is an isomorphism since it is an iterate of the Bott periodicity isomorphism.

Adams’ Hopf Invariant One Theorem Section 2.3 51

K D e ( 2n { }, })

»

£ D e @ 2n£{ K e D({ } 2n, } )

»

e D£ { £@ 2n

K D2nD2n D( , 2n )

»

£ D@ 2n£ K D2n D2n D( , 2n )

»

£ D2n£@ K D @ 2n D2n D( , ( 2n ))

»

£ D¡¡¡¡¡! 2n£

K C D( , 2n )

» ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡!

¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡!

f -

K C D( , 2n )

»

f K C S( , 2n )

»

+ f

K(C )

» ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡!

f K(C )

»

f K(C )

»

f

¡! ¡! ¡!

¼

¼ ¼

¡! ¡!

¼

­

­

­

­

­

ؤ ؤ ؤ

By the definition of an H–space and the definition of f , the map Ø restricts to a

homeomorphism from D2n_feg onto D2n

. and from feg_D2n onto D2n

− . It follows

that the element __ in the upper left group maps to a generator of the group in the

bottom row of the diagram, since _ restricts to a generator of e K.S2n. by definition.

Therefore by commutativity of the diagram, the product map in the top row sends

__ to __ since _ was defined to be the image of a generator of e K.Cf ; S2n.. Thus

we have _2 . __, which says that the Hopf invariant of f is _1. tu

In view of this lemma, Theorem 2.14 becomes a consequence of the following

theorem of Adams:

Theorem 2.17. If f : S4n−1!S2n is a map whose mod 2 Hopf invariant is 1, then

n . 1, 2, or 4.

The proof of this will occupy the rest of this section.

Adams Operations

The Hopf invariant is defined in terms of the ring structure in K–theory, but in

order to prove Adams’ theorem, more structure is needed, namely certain ring homomorphisms

k : K.X.!K.X.. Here are their basic properties:

Theorem 2.18. There exist ring homomorphisms k : K.X.!K.X., defined for all

compact Hausdorff spaces X and all integers k _ 0, and satisfying:

(1) kf _ . f _ k for all maps f :X!Y . (Naturality)

(2) k.L. . Lk if L is a line bundle.

(3) k _ ` . k` .

(4) p._. _ _p mod p for p prime.

This last statement means that p._. − _p . p_ for some _ 2 K.X..

In the special case of a vector bundle E which is a sum of line bundles Li , properties

(2) and (3) give the formula k.L1

_ ___ _Ln. . Lk1

. ___ . Lk

n . We would like

a general definition of k.E. which specializes to this formula when E is a sum of

line bundles. The idea is to use the exterior powers _k.E.. From the corresponding

properties for vector spaces we have:

52 Chapter 2 Complex K–Theory

(i) _k.E1

_E2. _

L

i

􀀀

_i.E1._k−i.E2.

_

.

(ii) _0.E. . 1, the trivial line bundle.

(iii) _1.E. . E.

(iv) _k.E. . 0 for k greater than the maximum dimension of the fibers of E.

Recall that we want k.E. to be Lk1

.___.Lk

n when E . L1

_ ___ _Ln for line bundles

Li . We will show in this case that there is a polynomial sk with integer coefficients

such that Lk1

. ___ . Lk

n

. sk._1.E.; ___ ; _k.E... This will lead us to define k.E. .

sk._1.E.; ___ ; _k.E.. for an arbitrary vector bundle E.

To see what the polynomial sk should be, we first use the exterior powers _i.E.

to define a polynomial _t.E. .

P

i _i.E.ti 2 K.X..t.. This is a finite sum by property

(iv), and property (i) says that _t.E1

_E2. . _t.E1._t.E2.. When E . L1

_ ___ _Ln

this implies that _t.E. .

Q

i_t.Li., which equals

Q

i.1 . Lit. by (ii), (iii), and (iv).

The coefficient _j.E. of tj in _t.E. .

Q

i.1 . Lit. is the jth elementary symmetric

function _j of the Li ’s, the sum of all products of j distinct Li ’s. Thus we have

._. _j.E. . _j.L1; ___ ; Ln. if E . L1

_ ___ _Ln

To make the discussion completely algebraic, let us introduce the variable ti for

Li . Thus .1.t1. ___ .1.tn. . 1._1 .___._n, where _j is the jth elementary symmetric

polynomial in the ti ’s. The fundamental theorem on symmetric polynomials,

proved for example in [Lang, p. 134] or [van der Waerden, x26], asserts that every

degree k symmetric polynomial in t1; ___ ; tn can be expressed as a unique polynomial

in _1; ___;_k . In particular, tk

1 .___.tkn

is a polynomial sk._1; ___;_k., called a

Newton polynomial. This polynomial sk is independent of n since we can pass from

n to n − 1 by setting tn

. 0. A recursive formula for sk is

sk

. _1sk−1 − _2sk−2 . ___..−1.k−2_k−1s1 ..−1.k−1k_k

To derive this we may take n . k, and then if we substitute x . −ti in the identity

.x . t1. ___ .x . tk. . xk . _1xk−1 . ___ . _k we get tk

i

. _1tk−1

i

− ___..−1.k−1_k .

Summing over i then gives the recursion relation. The recursion relation easily yields

for example

s1 . _1 s2 . _2

1 − 2_2 s3 . _3

1 − 3_1_2 . 3_3

s4 . _4

1 − 4_2

1 _2 . 4_1_3 . 2_2

2 − 4_4

Summarizing, if we define k.E. . sk._1.E.; ___ ; _k.E.., then in the case that E

is a sum of line bundles L1

_ ___ _Ln we have

k.E. . sk._1.E.; ___ ; _k.E..

. sk._1.L1; ___ ; Ln.; ___;_k.L1; ___ ; Ln.. by ._.

. Lk1

. ___.Lk

n

Verifying that the definition k.E. . sk._1.E.; ___ ; _k.E.. gives operations on

K.X. satisfying the properties listed in the theorem will be rather easy if we make

use of the following general result:

Adams’ Hopf Invariant One Theorem Section 2.3 53

The Splitting Principle. Given a vector bundle E!X with X compact Hausdorff,

there is a compact Hausdorff space F.E. and a map p : F.E.!X such that the induced

map p_ :K_.X.!K_.F.E.. is injective and p_.E. splits as a sum of line

bundles.

This will be proved later in this section, but for the moment let us assume it and

proceed with the proof of Theorem 2.18 and Adams’ theorem.

Proof of Theorem 2.18: Property (1) for vector bundles, f _. k.E.. . k.f _.E..,

follows immediately from the relation f _._i.E.. . _i.f _.E... Additivity of k for

vector bundles, k.E1

_E2. . k.E1. . k.E2., follows from the splitting principle

since we can first pull back to split E1 then do a further pullback to split E2 , and the

formula k.L1

_ ___ _Ln. . Lk1

. ___.Lk

n preserves sums. Since k is additive on

vector bundles, it induces an additive operation on K.X. defined by k.E1 − E2. .

k.E1. − k.E2..

For this extended k the properties (1) and (2) are clear. Multiplicativity is also

easy from the splitting principle: If E is the sum of line bundles Li and E0 is the sum

of line bundles L0

j , then EE0 is the sum of the line bundles Li

L0

j , so k.EE0. .

P

i;j k.Li

L0

j. .

P

i;j.Li

L0

j.k .

P

i;j Lki

L0

j

k .

P

i Lki

P

j L0

j

k . k.E. k.E0.. Thus

k is multiplicative for vector bundles, and it follows formally that it is multiplicative

on elements of K.X.. For property (3) the splitting principle and additivity reduce

us to the case of line bundles, where k. `.L.. . Lk` . k`.L.. Likewise for (4), if

E . L1 . ___.Ln, then p.E. . Lp1

. ___.Lp

n

_ .L1 . ___.Ln.p . Ep mod p. tu

By the naturality property (1), k restricts to an operation k : e K.X.!e K.X. since

e K.X. is the kernel of the homomorphism K.X.!K.x0. for x0 2 X. For the external

product e K.X. e K.Y.!e K.X ^ Y., we have the formula k._ _ _. . k._. _ k._.

since if one looks back at the definition of _ _ _, one sees this was defined as

p_

1 ._.p_

2 ._., hence

k._ _ _. . k.p_

1 ._.p_

2 ._..

. k.p_

1 ._.. k.p_

2 ._..

. p_

1 . k._..p_

2 . k._..

. k._. _ k._.:

This will allow us to compute k on e K.S2n. _ Z. In this case k must be

multiplication by some integer since it is an additive homomorphism of Z.

Proposition 2.19. k : e K.S2n.!e K.S2n. is multiplication by kn .

Proof: Consider first the case n . 1. Since k is additive, it will suffice to show

k._. . k_ for _ a generator of e K.S2.. We can take _ . H − 1 for H the canonical

54 Chapter 2 Complex K–Theory

line bundle over S2 . CP1 . Then

k._. . k.H − 1. . Hk − 1 by property (2)

. .1 . _.k − 1

. 1 . k_ − 1 since _i . .H − 1.i . 0 for i _ 2

. k_

When n > 1 we use the external product e K.S2. e K.S2n−2.!e K.S2n., which is

an isomorphism, and argue by induction. Assuming the desired formula holds in

e K.S2n−2., we have k._ _ _. . k._. _ k._. . k_ _ kn−1_ . kn._ _ _., and we

are done. tu

Now we can use the operations 2 and 3 and the relation 2 3 . 6 . 3 2

to prove Adams’ theorem.

Proof of Theorem 2.17: The definition of the Hopf invariant of a map f : S4n−1!S2n

involved elements _;_ 2 e K.Cf .. By Proposition 2.19, k._. . k2n_ since _ is the

image of a generator of e K.S4n.. Similarly, k._. . kn_ . _k_ for some _k

2 Z.

Therefore

k `._. . k.`n_ . _`_. . kn`n_ . .k2n_`

. `n_k._

Since k ` . k` . ` k , the coefficient k2n_`

. `n_k of _ is unchanged when k

and ` are switched. This gives the relation

k2n_`

. `n_k

. `2n_k

. kn_`; or .k2n − kn._`

. .`2n − `n._k

By property (6) of 2 , we have 2._. _ _2 mod 2. Since _2 . h_ with h the Hopf

invariant of f , the formula 2._. . 2n_ . _2_ implies that _2 _ h mod 2, so _2 is

odd if we assume h . _1. By the preceding displayed formula we have .22n−2n._3 .

.32n − 3n._2, or 2n.2n−1._3 . 3n.3n − 1._2, so 2n divides 3n.3n − 1._2 . Since 3n

and _2 are odd, 2n must then divide 3n −1. The proof is completed by the following

elementary number theory fact. tu

Lemma 2.20. If 2n divides 3n − 1 then n . 1; 2, or 4.

Proof: Write n . 2`m with m odd. We will show that the highest power of 2 dividing

3n − 1 is 2 for `.0 and 2`.2 for ` > 0. This implies the lemma since if 2n divides

3n −1, then by this fact, n _ `.2, hence 2` _ 2`m . n _ `.2, which implies ` _ 2

and n _ 4. The cases n . 1; 2; 3; 4 can then be checked individually.

We find the highest power of 2 dividing 3n − 1 by induction on `. For ` . 0

we have 3n − 1 . 3m − 1 _ 2 mod 4 since 3 _ −1 mod 4 and m is odd. In the next

case ` . 1 we have 3n − 1 . 32m − 1 . .3m − 1..3m . 1.. The highest power of 2

dividing the first factor is 2 as we just showed, and the highest power of 2 dividing

the second factor is 4 since 3m . 1 _ 4 mod 8 because 32 _ 1 mod 8 and m is

Adams’ Hopf Invariant One Theorem Section 2.3 55

odd. So the highest power of 2 dividing the product .3m − 1..3m . 1. is 8. For the

inductive step of passing from ` to ` . 1 with ` _ 1, or in other words from n to

2n with n even, write 32n − 1 . .3n − 1..3n . 1.. Then 3n . 1 _ 2 mod 4 since n

is even, so the highest power of 2 dividing 3n . 1 is 2. Thus the highest power of 2

dividing 32n − 1 is twice the highest power of 2 dividing 3n − 1. tu

The Splitting Principle

The splitting principal will be a fairly easy consequence of a general result about

the K–theory of fiber bundles called the Leray-Hirsch theorem, together with a calculation

of the ring structure of K_.CPn.. The following proposition will allow us to

compute at least the additive structure of K_.CPn..

Proposition 2.21. If X is a finite cell complex with n cells, then K_.X. is a finitely

generated group with at most n generators. If all the cells of X have even dimension

then K1.X. . 0 and K0.X. is free abelian with one basis element for each cell.

The phrase ‘finite cell complex’ would normally mean ‘finite CW complex’ but we

can take it to be something slightly more general: a space built from a finite discrete set

by attaching a finite number of cells in succession, with no conditions on the dimensions

of these cells, so cells are not required to attach only to cells of lower dimension.

Finite cell complexes are always homotopy equivalent to finite CW complexes (by deforming

each succesive attaching map to be cellular) so the only advantages of finite

cell complexes are technical. In particular, it is easy to see that a space is a finite cell

complex if it is a fiber bundle over a finite cell complex with fibers that are also finite

cell complexes. This is shown in Proposition 2.26 in a brief appendix to this section.

It implies that the splitting principal can be applied staying within the realm of finite

cell complexes.

Proof: We show this by induction on the number of cells. The complex X is obtained

from a subcomplex A by attaching a k cell, for some k. For the pair .X;A. we

have an exact sequence eK

_.X=A. -! eK

_.X. -! eK

_.A.. Since X=A . Sk , we have

eK

_.X=A. _ Z, and exactness implies that eK

_.X. requires at most one more generator

than eK

_.A..

The first term of the exact sequence K1.X=A.!K1.X.!K1.A. is zero if all

cells of X are of even dimension, so induction on the number of cells implies that

K1.X. . 0. Then there is a short exact sequence 0!eK

0.X=A.!eK

0.X.!eK

0.A.!0

with eK

0.X=A. _ Z. By induction eK

.A. is free, so this sequence splits, hence K0.X. _

Z_K0.A. and the final statement of the proposition follows. tu

This proposition applies in particular to CPn , which has a cell structure with one

cell in each dimenion 0; 2; 4; ___ ; 2n, so K1.CPn. . 0 and K0.CPn. _ Zn.1 . The ring

structure is as simple as one could hope for:

56 Chapter 2 Complex K–Theory

Proposition 2.22. K.CPn. is the quotient ring Z.L.=.L−1.n.1 where L is the canonical

line bundle over CPn .

Thus by the change of variable x . L−1 we see that K.CPn. is the truncated polynomial

ring Z.x.=.xn.1., with additive basis 1;x; ___;xn. It follows that 1; L; ___ ; Ln

is also an additive basis.

Proof: The exact sequence for the pair .CPn;CPn−1. gives a short exact sequence

0 -!K.CPn;CPn−1.!- K.CPn. --!-_ K.CPn−1.!- 0

We shall prove:

.an. .L − 1.n generates the kernel of the restriction map _.

Hence if we assume inductively that K.CPn−1. . Z.L.=.L − 1.n , then .an. and the

preceding exact sequence imply that f1; L − 1; ___ ; .L − 1.ng is an additive basis for

K.CPn.. Since .L − 1.n.1 . 0 in K.CPn. by .an.1., it follows that K.CPn. is the

quotient ring Z.L.=.L − 1.n.1 , completing the induction.

A reason one might expect .an. to be true is that the kernel of _ can be identified

with K.CPn;CPn−1. . e K.S2n. by the short exact sequence, and Bott periodicity

implies that the n fold reduced external product of the generator L − 1 of eK.S2.

with itself generates e K.S2n.. To make this rough argument into a proof we will have

to relate the external product e K.S2. ___  e K.S2.!e K.S2n. to the ‘internal’ product

K.CPn. ___ K.CPn.!K.CPn..

The space CPn is the quotient of the unit sphere S2n.1 in Cn.1 under multiplication

by scalars in S1 _ C. Instead of S2n.1 we could equally well take the boundary

of the product D2

0_ ___ _D2n

where D2

i is the unit disk in the ith coordinate of Cn.1 ,

and we start the count with i . 0 for convenience. Then we have

@.D2

0_ ___ _D2n

. .

S

i.D2

0_ ____@D2

i

_ ____D2n

.

The action of S1 by scalar multiplication respects this decomposition. The orbit space

of D2

0_ ____@D2

i

_ ____D2n

under the action is a subspace Ci

_ CPn homeomorphic

to the product D2

0_ ____D2n

with the factor D2

i deleted. Thus we have a decomposition

CPn .

S

i Ci with each Ci homeomorphic to D2n and with Ci

\ Cj

. @Ci

\ @Cj

for i . j .

Consider now C0 . D2

1_ ____D2n

. Its boundary is decomposed into the pieces

@iC0 . D2

1_ ____@D2

i

_ ____D2n

. The inclusions .D2

i ; @D2

i . _ .C0; @iC0. _ .CPn; Ci.

give rise to a commutative diagram

Adams’ Hopf Invariant One Theorem Section 2.3 57

K D2

1 D2

1 ( , @ ) K D2

nD2

n ( , @ )

¡¡¡¡¡¡¡¡¡¡!

¡¡¡¡¡! ¡¡¡¡¡!

¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡!

¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡!

¡¡¡!

¡¡!

¡!

¡! ¡!

¡!

¼

¼

. . .

K C0 1C0 0 0 ( , @ ) . . . K(C ,@nC ) K(C0 ,@C0 )

K( P ,C1) n

n . . . . . .

¼ ¼

¼

C K C P ( ) K 1 C ( P , n

, C n C ) ) C n

K(CPn ) . . . K(CPn ) K(CPn )

[ [ K( P , C n CP n - 1

­ ­

­ ­

­ ­

­ ­

where the maps from the first column to the second are the n fold products. The

upper map in the middle column is an isomorphism because the inclusion C0>CPn

induces a homeomorphism C0=@C0 _ CPn=.C1[___[Cn.. The CPn−1 at the right side

of the diagram sits in CPn in the last n coordinates of Cn.1 , so is disjoint from C0 ,

hence the quotient map CPn=CPn−1!CPn=.C1[___[Cn. is a homotopy equivalence.

The element xi

2 K.CPn; Ci. mapping downward to L−1 2 K.CPn. maps upward

to a generator of K.C0; @iC0. _ K.D2

i ; @D2

i .. By commutativity of the diagram, the

product x1 ___xn then generates K.CPn; C1 [ ___ [ Cn.. This means that .L − 1.n

generates the image of the map K.CPn;CPn−1.!K.CPn., which equals the kernel of

_, proving .an.. tu

Here is a version of the Leray-Hirsch theorem for K–theory:

Theorem 2.23. Let p : E!B be a fiber bundle with E and B compact Hausdorff and

with fiber F such that K_.F. is free. Suppose that there exist classes c1; ___ ; ck

2

K_.E. that restrict to a basis for K_.F. in each fiber F . If either

(a) B is a finite cell complex, or

(b) F is a finite cell complex having all cells of even dimension,

then K_.E., as a module over K_.B., is free with basis fc1; ___ ; ck

g.

Here the K_.B. module structure on K_.E. is defined by _ _  . p_._. for

_ 2 K_.B. and  2 K_.E.. Another way to state the conclusion of the theorem is

to say that the map Ø :K_.B.K_.F.!K_.E., Ø.

P

i bi i_.ci.. .

P

i p_.bi.ci for i

the inclusion F>E, is an isomorphism.

In the case of the product bundle E . F_B the classes ci can be chosen to be the

pullbacks under the projection E!F of a basis for K_.F.. The theorem then asserts

that the external product K_.F.K_.B.!K_.F_B. is an isomorphism.

For most of our applications of the theorem either case (a) or case (b) will suffice.

The proof of (a) is somewhat simpler than (b), and we include (b) mainly to obtain the

splitting principle for vector bundles over arbitrary compact Hausdorff base spaces.

Proof: For a subspace B0 _ B let E0 . p−1.B0.. Then we have a diagram

58 Chapter 2 Complex K–Theory

¡¡!

¡¡!

¡¡!

¡¡¡¡¡! ¤ 0 ¡¡¡¡¡!

¤

­

Ø Ø Ø

K (B B )

( )

, ¤ K (F ) ¤ ¡¡¡¡¡! K (B ) ­ ¤ K (F ) ¤ 0 ¡¡¡! K (B ) ­ ¤K (F )

¡¡¡¡¡¡¡¡¡¡¡¡! ¤ 0 ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡! K (E,E ) ¤ ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡! K (E ) ¤ 0 ¡¡¡¡¡¡¡¡! K (E )

where the left-hand Ø is defined by the same formula Ø.

P

i bi i_.ci.. .

P

i p_.bi.ci ,

but with p_.bi.ci referring now to the relative product K_.E; E0._K_.E.!K_.E; E0..

The right-hand Ø is defined using the restrictions of the ci ’s to the subspace E0. To

see that the diagram ._. commutes, we can interpolate between its two rows the row

-!K_.E; E0.K_.F. -!K_.E.K_.F. -!K_.E0.K_.F. -!

by factoring Ø as the composition

P

i bi i_.ci.,P

i p_.bi.i_.ci.,P

i p_.bi.ci .

The upper squares of the enlarged diagram then commute trivially, and the lower

squares commute by Proposition 2.8. The lower row of the diagram is of course exact.

The upper row is also exact since we assume K_.F. is free, and tensoring an exact

sequence with a free abelian group preserves exactness, the result of the tensoring

operation being simply to replace the given exact sequence by the direct sum of a

number of copies of itself.

The proof in case (a) will be by a double induction, first on the dimension of B,

then within a given dimension, on the number of cells. The induction starts with the

trivial case that B is zero-dimensional, hence a finite discrete set. For the induction

step, suppose B is obtained from a subcomplex B0 by attaching a cell en , and let

E0 . p−1.B0. as above. By induction on the number of cells of B we may assume the

right-hand Ø in ._. is an isomorphism. If the left-hand Ø is also an isomorphism,

then the five-lemma will imply that the middle Ø is an isomorphism, finishing the

induction step.

Let ': .Dn; Sn−1.!.B; B0. be a characteristic map for the attached n cell. Since

Dn is contractible, the pullback bundle '_.E. is a product, and so we have a commutative

diagram

¡¡!

¡¡!

¤ 0 ¡¡¡¡¡! ­

Ø Ø Ø

K (B,B ) , ¤ K (F ) ­ ¤ K (F ) ¤ K (D S )

¤ 0 ¡¡¡¡¡¡¡¡¡¡¡¡! 0 K (E,E ) , ¤ ¤ K ( (E )

¡¡¡¡!

¤ K (

¼

¼ ' ¼ ¤ ' (E )) £ £

n n - 1

Dn F,S n - 1 F )

The two horizontal maps are isomorphisms since ' restricts to a homeomorphism

on the interior of Dn , hence induces homeomorphisms B=B0 _ Dn=Sn−1 and E=E0 _

'_.E.='_.E0.. Thus the diagram reduces the proof to showing that the right-hand

Ø, involving the product bundle Dn_F!Dn , is an isomorphism.

Consider the diagram ._. with .B; B0. replaced by .Dn; Sn−1.. We may assume

the right-hand Ø in ._. is an isomorphism since Sn−1 has smaller dimension than

the original cell complex B. The middle Ø is an isomorphism by the case of zerodimensional

B since Dn deformation retracts to a point. Therefore by the five-lemma

Adams’ Hopf Invariant One Theorem Section 2.3 59

the left-hand Ø in ._. is an isomorphism for .B; B0. . .Dn; Sn−1.. This finishes the

proof in case (a).

In case (b) let us first prove the result for a product bundle E . F_B. In this case

Ù is just the external product, so we are free to interchange the roles of F and B.

Thus we may use the diagram ._. with F an arbitrary compact Hausdorff space and

B a finite cell complex having all cells of even dimension, obtained by attaching a cell

en to a subcomplex B0 . The upper row of ._. is then an exact sequence since it is

obtained from the split short exact sequence 0!K_.B; B0.!K_.B.!K_.B0.!0 by

tensoring with the fixed group K_.F.. If we can show that the left-hand Ø in ._. is

an isomorphism, then by induction on the number of cells of B we may assume the

right-hand Ø is an isomorphism, so the five-lemma will imply that the middle Ø is

also an isomorphism.

To show the left-hand Ø is an isomorphism, note first that B=B0 . Sn so we may

as well take the pair .B; B0. to be .Dn; Sn−1.. Then the middle Ø in ._. is obviously

an isomorphism, so the left-hand Ø will be an isomorphism iff the right-hand Ø is

an isomorphism. When the sphere Sn−1 is even-dimensional we have already shown

that Ø is an isomorphism in the remarks following the proof of Lemma 2.15, and

the same argument applies also when the sphere is odd-dimensional, since K1 of an

odd-dimensional sphere is K0 of an even-dimensional sphere.

Now we turn to case (b) for nonproducts. The proof will once again be inductive,

but this time we need a more subtle inductive statement since B is just a compact

Hausdorff space, not a cell complex. Consider the following condition on a compact

subspace U _ B:

For all compact V _ U the map Ø :K_.V.K_.F.!K_.p−1.V.. is an isomorphism.

If this is satisfied, let us call U good. By the special case already proved, each point of

B has a compact neighborhood U that is good. Since B is compact, a finite number

of these neighborhoods cover B, so by induction it will be enough to show that if U1

and U2 are good, then so is U1 [ U2 .

A compact V _ U1 [ U2 is the union of V1 . V \ U1 and V2 . V \ U2 . Consider

the diagram like ._. for the pair .V;V2.. Since K_.F. is free, the upper row of this

diagram is exact. Assuming U2 is good, the map Ø is an isomorphism for V2, so Ø

will be an isomorphism for V if it is an isomorphism for .V;V2.. The quotient V=V2

is homeomorphic to V1=.V1 \ V2. so Ø will be an isomorphism for .V;V2. if it is an

isomorphism for .V1; V1 \ V2.. Now look at the diagram like ._. for .V1; V1 \ V2..

Assuming U1 is good, the maps Ø are isomorphisms for V1 and V1 \V2 . Hence Ø is

an isomorphism for .V1; V1 \V2., and the induction step is finished. tu

Example 2.24. Let E!X be a vector bundle with fibers Cn and compact base X.

Then we have an associated projective bundle p : P.E.!X with fibers CPn−1 , where

60 Chapter 2 Complex K–Theory

P.E. is the space of lines in E, that is, one-dimensional linear subspaces of fibers of

E. Over P.E. there is the canonical line bundle L!P.E. consisting of the vectors in

the lines of P.E.. In each fiber CPn−1 of P.E. the classes 1; L; ___ ; Ln−1 in K_.P.E..

restrict to a basis for K_.CPn−1. by Proposition 2.22. From the Leray-Hirsch theorem

we deduce that K_.P.E.. is a free K_.X. module with basis 1; L; ___ ; Ln−1 .

Proof of the Splitting Principle: In the preceding example, the fact that 1 is among the

basis elements implies that p_ :K_.X.!K_.P.E.. is injective. The pullback bundle

p_.E.!P.E. contains the line bundle L as a subbundle, hence splits as L_E0 for

E0!P.E. the subbundle of p_.E. orthogonal to L with respect to some choice of

inner product. Now repeat the process by forming P.E0., splitting off another line

bundle from the pullback of E0 over P.E0.. Note that P.E0. is the space of pairs

of orthogonal lines in fibers of E. After a finite number of repetitions we obtain

the flag bundle F.E.!X described at the end of x1.1, whose points are n tuples of

orthogonal lines in fibers of E, where n is the dimension of E. (If the fibers of E

have different dimensions over different components of X, we do the construction

for each component separately.) The pullback of E over F.E. splits as a sum of line

bundles, and the map F.E.!X induces an injection on K_ since it is a composition

of maps with this property. tu

In the preceding Example 2.24 we saw that K_.P.E.. is free as a K_.X. module,

with basis 1; L; ___ ; Ln−1 . In order to describe the multiplication in K_.P.E.. one

therefore needs only a relation expressing Ln in terms of lower powers of L. Such

a relation can be found as follows. The pullback of E over P.E. splits as L_E0 for

some bundle E0 of dimension n − 1, and the desired relation will be _n.E0. . 0. To

compute _n.E0. . 0 we use the formula _t.E. . _t.L._t.E0. in K_.P.E...t., where

to simplify notation we let ‘E’ also denote the pullback of E over P.E.. The equation

_t.E. . _t.L._t.E0. can be rewritten as _t.E0. . _t.E._t.L.−1 where _t.L.−1 .

P

i.−1.iLiti since _t.L. . 1 . Lt . Equating coefficients of tn in the two sides of

_t.E0. . _t.E._t.L.−1 , we get _n.E0. .

P

i.−1.n−i_i.E.Ln−i . The relation _n.E0. . 0

can be written as

P

i.−1.i_i.E.Ln−i . 0, with the coefficient of Ln equal to 1, as

desired. The result can be stated in the following form:

Proposition 2.25. For an n dimensional vector bundle E!X the ring K.P.E.. is

isomorphic to the quotient ring K_.X..L.=

􀀀P

i.−1.i_i.E.Ln−i_

. tu

For example when X is a point we have P.E. . CPn−1 and _i.E. . Ck for k .

_

ni

_

,

so the polynomial

P

i.−1.i_i.E.Ln−i becomes .L−1.n and we see that the proposition

generalizes the isomorphism K_.CPn−1. _ Z.L.=.L − 1.n..

Appendix: Finite Cell Complexes

As we mentioned in the remarks following Proposition 2.21 it is convenient for

purposes of the splitting principal to work with spaces slightly more general than finite

Further Calculations Section 2.4 61

CW complexes. By a finite cell complex we mean a space which has a finite filtration

X0 _ X1 _ ___ _ Xk

. X where X0 is a finite discrete set and Xi.1 is obtained from

Xi by attaching a cell eni via a map 'i : Sni−1!Xi . Thus Xi.1 is the quotient space

of the disjoint union of Xi and a disk Dni under the identifications x _ 'i.x. for

x 2 @Dni . Sni−1 .

Proposition 2.26. If p : E!B is a fiber bundle whose fiber F and base B are both

finite cell complexes, then E is also a finite cell complex, whose cells are products of

cells in B with cells in F .

Proof: Suppose B is obtained from a subcomplex B0 by attaching a cell en . By induction

on the number of cells of B we may assume that p−1.B0. is a finite cell complex.

If Ø :Dn!B is a characteristic map for en then the pullback bundle Ø_.E.!Dn is

a product since Dn is contractible. Since F is a finite cell complex, this means that

we may obtain Ø_.E. from its restriction over Sn−1 by attaching cells. Hence we may

obtain E from p−1.B0. by attaching cells. tu

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