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3.2. The Chern Character

Back

In this section we apply the most basic facts about Chern classes to obtain a direct

connection between K–theory and ordinary cohomology. This is then used to study

the J–homomorphism, which maps the homotopy groups of orthogonal and unitary

groups to the homotopy groups of spheres.

The total Chern class c . 1 . c1 . c2 . ___ takes direct sums to cup products,

and the idea of the Chern character is to form an algebraic combination of Chern

classes which takes direct sums to sums and tensor products to cup products, thus

giving a natural ring homomorphism from K–theory to cohomology. In order to make

this work one must use cohomology with rational coefficients, however. The situation

might have been simpler if it had been possible to use integer coefficients instead, but

on the other hand, the fact that one has rational coefficients instead of integers make

it possible to define a homomorphism e :_2m−1.S2n.!Q=Z which gives some very

interesting information about the difficult subject of homotopy groups of spheres.

In order to define the Chern character it suffices, via the splitting principle, to do

the case of line bundles. The idea is to define the Chern character ch.L. for a line

74 Chapter 3 Characteristic Classes

bundle L!X to be ch.L. . ec1.L. . 1 . c1.L. . c1.L.2=2! . ___ 2 H_.X;Q., so that

ch.L1

L2. . ec1.L1L2. . ec1.L1..c1.L2. . ec1.L1.ec1.L2. . ch.L1.ch.L2.. If the sum

1.c1.L..c1.L.2=2!.___ has infinitely many nonzero terms, it will lie not in the direct

sum H_.X;Q. of the groups Hn.X;Q. but rather in the direct product. However, in

the examples we shall be considering, Hn.X;Q. will be zero for sufficiently large n,

so this distinction will not matter.

For a direct sum of line bundles E _ L1

_ ___ _Ln we would then want to have

ch.E. .

ch.Li. .

eti . n . .t1 . ___ . tn. . ___..tk

1 . ___ . tkn

.=k! . ___

where ti

. c1.Li.. The total Chern class c.E. is then .1 . t1. ___ .1 . tn. . 1 . _1 .

___._n , where _j

. cj.E. is the jth elementary symmetric polynomial in the ti ’s, the

sum of all products of j distinct ti ’s. As we saw in x2.3, the Newton polynomials sk

satisfy tk

1 .___.tkn

. sk._1; ___;_k.. Since _j

. cj.E., this means that the preceding

displayed formula can be rewritten

ch.E. . dim E .

k>0

sk.c1.E.; ___ ; ck.E..=k!

The right side of this equation is defined for arbitrary vector bundles E, so we take

this as our general definition of ch.E..

Proposition 3.12. ch.E1

_E2. . ch.E1..ch.E2. and ch.E1

E2. . ch.E1.ch.E2..

Proof: The proof of the splitting principle for ordinary cohomology in Proposition

2.3 works with any coefficients in the case of complex vector bundles, in particular

for Q coefficients. By this splitting principle we can pull E1 back to a sum of line

bundles over a space F.E1.. By another application of the splitting principle to the

pullback of E2 over F.E1., we have a map F.E1; E2.!X pulling both E1 and E2 back

to sums of line bundles, with the induced map H_.X;Q.!H_.F.E1; E2.;Q. injective.

So to prove the proposition it suffices to verify the two formulas when E1 and E2

are sums of line bundles, say Ei

. _

jLij for i . 1; 2. The sum formula holds since

ch.E1

_E2. . ch._

i;jLij. .

i;j ec1.Lij. . ch.E1..ch.E2., by the discussion preceding

the definition of ch. For the product formula, ch.E1

E2. . ch

􀀀

j;k.L1j

L2k.

j;k ch.L1j

L2k. .

j;k ch.L1j.ch.L2k. . ch.E1.ch.E2.. tu

In view of this proposition, the Chern character automatically extends to a ring

homomorphism ch: K.X.!H_.X;Q.. By naturality there is also a reduced form

ch: e K.X.!eH

_.X;Q. since these reduced rings are the kernels of restriction to a

point.

As a first calculation of the Chern character, we have:

Proposition 3.13. ch: e K.S2n.!H2n.S2n;Q. is injective with image equal to the

subgroup H2n.S2n; Z. _ H2n.S2n;Q..

Stiefel-Whitney and Chern Classes Section 3.2 75

Proof: Since ch.x .H − 1.. . ch.x. ` ch.H − 1. we have the commutative diagram

shown at the right, where the upper map is external

¡!

¡¡¡¡¡!

¡¡!

K(X )

H (X; )

e ¼

Q ¤

¡!

K S2X

S X 2

( )

H ( ; )

e Q ¤

tensor product with H − 1, which is an isomorphism by

Bott periodicity, and the lower map is cross product with

ch.H − 1. . ch.H. − ch.1. . 1 . c1.H. − 1 . c1.H., a

generator of H2.S2; Z.. From Theorem 3.16 of [AT] the lower map is an isomorphism

and restricts to an isomorphism of the Z coefficient subgroups. Taking X . S2n , the

result now follows by induction on n, starting with the trivial case n . 0. tu

An interesting by-product of this is:

Corollary 3.14. A class in H2n.S2n; Z. occurs as a Chern class cn.E. iff it is divisible

by .n − 1.! .

Proof: For vector bundles E!S2n we have c1.E. . ___ . cn−1.E. . 0, so ch.E. .

dim E.sn.c1; ___ ; cn.=n! . dim E_ncn.E.=n! by the recursion relation for sn derived

in x2.3, namely, sn

. _1sn−1 − _2sn−2 . ___..−1.n−2_n−1s1 ..−1.n−1n_n . tu

Even when H_.X; Z. is torsionfree, so that H_.X; Z. is a subring of H_.X;Q.,

it is not always true that the image of ch is contained in H_.X; Z.. For example, if

L 2 K.CPn. is the canonical line bundle, then ch.L. . 1.c.c2=2.___.cn=n! where

c . c1.L. generates H2.CPn; Z., hence ck generates H2k.CPn; Z. for k _ n.

The Chern character can be used to show that for finite cell complexes X, the

only possible differences between the groups K_.X. and H_.X; Z. lie in their torsion

subgroups. Since these are finitely generated abelian groups, this will follow if we can

show that K_.X.Q and H_.X;Q. are isomorphic.

Proposition 3.15. The map K_.X.Q!H_.X;Q. induced by the Chern character

is an isomorphism for all finite cell complexes X.

Proof: We proceed by induction on the number of cells of X. The result is trivially

true when there is a single cell, a 0 cell, and it is also true when there are

two cells, so that X is a sphere, by the preceding proposition. For the induction

step, let X be obtained from a subcomplex A by attaching a cell. Consider the fiveterm

sequence X=A!SA!SX!SX=SA!S2A. Applying the rationalized Chern

character K_.−.Q!H_.−;Q. then gives a commutative diagram of five-term exact

sequences since tensoring with Q preserves exactness. The space X=A is a

sphere, and SX=SA is homotopy equivalent to a sphere. Both SA and S2A are homotopy

equivalent to cell complexes with the same number of cells as A, by collapsing

the suspension or double suspension of a 0 cell. Thus by induction four of the

five maps between the two exact sequences are isomorphisms, all except the map

K_.SX.Q!H_.SX;Q., so by the five-lemma this map is an isomorphism as well.

76 Chapter 3 Characteristic Classes

Finally, to obtain the result for X itself we may replace X by S2X since the Chern character

commutes with double suspension, as we have seen, and a double suspension

is in particular a single suspension, with the same number of cells, up to homotopy

equivalence. tu

The J–Homomorphism

Homotopy groups of spheres are notoriously difficult to compute, but some partial

information can be gleaned from certain naturally defined homomorphisms

J :_i.O.n..!_n.i.Sn.

One of the goals of this book is to determine these J homomorphisms in the stable

dimension range n >> i where both domain and range are independent of n, according

to Proposition 1.14 for O.n. and the Freudenthal suspension theorem [AT] for

Sn . The real form of Bott periodicity proved in Chapter 4 implies that the domain of

the stable J homomorphism _i.O.!_s

i is nonzero only for i . 4n−1 when _i.O.

is Z and for i . 8n and 8n . 1 when _i.O. is Z2 . In the latter two cases we will

show in Chapter 4 that J is injective. When i . 4n−1 the image of J is a finite cyclic

group of some order an since _s

i is a finite group for i > 0 by a theorem of Serre

proved in [SSAT].

The values of an have been computed in terms of Bernouilli numbers. Here is a

table for small values of n:

¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡

¡¡¡¡¡¡¡¡¡¡¡¡

24 240 504 480 264 65520 24 16320 28728 13200 552

n 1 2 3 4 5 6 7 8 9 10 11

In spite of appearances, there is great regularity in this sequence, but this becomes

clear only when one looks at the prime factorization of an . Here are the rules for

computing an :

1. The highest power of 2 dividing an is 2`.3 where 2` is the highest power of 2

dividing n.

2. An odd prime p divides an iff n is a multiple of .p − 1.=2, and in this case

the highest power of p dividing an is p`.1 where p` is the highest power of p

dividing n.

The first three cases p . 2; 3; 5 are shown in the following diagram, where a vertical

chain of k connected dots above the number 4n−1 means that the highest power of

p dividing an is pk .

Stiefel-Whitney and Chern Classes Section 3.2 77

¡¡¡¡¡

3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 9911 95 99 103 107

¡p¡¡=¡2¡:

p = 3 :

¡p¡¡=¡5¡:

39 79 119 159 199 239 279 319 359 399 439 479 519 559 599 639 679 719 759 799 839 879 919 959 999

In the present section we will use the Chern character to show that an=2 is a lower

bound on the order of the image of J in dimension 4n − 1. Improving this bound to

an will be done in Chapter 4 using real K–theory. In Chapter ?? we will show that an

is also an upper bound for the order.

The simplest definition of the J homomorphism goes as follows. An element

.f . 2 _i.O.n.. is represented by a family of isometries fx

2 O.n., x 2 Si , with fx

the identity when x is the basepoint of Si . Writing Sn.i as @.Di.1_Dn. . Si_Dn [

Di.1_Sn−1 and Sn as Dn=@Dn , let Jf .x;y. . fx.y. for .x;y. 2 Si_Dn and let

Jf.Di.1_Sn−1. . @Dn , the basepoint of Dn=@Dn . Clearly f ' g implies Jf ' Jg,

so we have a map J :_i.O.n..!_n.i.Sn.. We will tacitly exclude the trivial case

i . 0.

Proposition 3.16. J is a homomorphism.

Proof: We can view Jf as a map In.i!Sn . Dn=@Dn which on Si_Dn _ In.1 is

given by .x;v.,fx.v. and which sends the complement of Si_Dn to the basepoint

@Dn . Taking a similar view of Jg, the sum Jf .Jg is obtained by juxtaposing these

two maps on either side of a hyperplane. We may assume fx is the identity for x in

the right half of Si and gx is the identity for x in the left half of Si . Then we obtain

a homotopy from Jf . Jg to J.f . g. by moving the two Si_Dn ’s together until

they coincide, as shown in the figure below. tu

¡¡! ¡¡!

We know that _i.O.n.. and _n.i.Sn. are independent of n for n > i.1, so

we would expect the J–homomorphism defined above

to induce a stable J–homomorphism J :_i.O.!_s

i , via

¡!

¡¡¡¡!

¡¡!

(O n))

(S ) S

¡!

¼i

( O ( ) ) ¼i(n +1

n +i

n +1 ¼n +i+1(S )

commutativity of the diagram at the right. We leave it as

an exercise for the reader to verify that this is the case.

78 Chapter 3 Characteristic Classes

Composing the stable J–homomorphism with the map _i.U.!_i.O. induced by

the natural inclusions U.n. _ O.2n. which give an inclusion U _ O, we get the stable

complex J–homomorphism JC :_i.U.!_s

i . Our goal is to define via K–theory a homomorphism

e :_s

i!Q=Z for i odd and compute the composition eJC :_i.U.!Q=Z.

This will give a lower bound for the order of the image of the real J–homomorphism

_i.O.!_s

i when i . 4n − 1.

Now let us define the main object we will be studying in this section, the homomorphism

e :_2m−1.S2n.!Q=Z. For a map f : S2m−1!S2n we have the mapping

cone Cf obtained by attaching a cell e2m to S2n by f . The quotient Cf =S2n is S2m

so we have a commutative diagram of short exact sequences

¡!

¡¡¡¡¡! ¡¡¡¡¡! ¡¡¡¡¡! ¡¡¡¡¡!

¡¡! ¡¡! ¡¡! ¡¡!

K(C )

0 H (C ; ) 0

e Q ¤

¡!

0 K S 0

( )

H ( ; )

e Q ¤

¡!

K S

( )

H ( ; )

e Q ¤

There are elements _;_ 2 e K.Cf . mapping from and to the standard generators

.H − 1. _ ___ _ .H − 1. of e K.S2m. and e K.S2n., respectively. In a similar way there

are elements a; b 2 eH

_.Cf ;Q. mapping from and to generators of H2m.S2m; Z. and

H2n.S2n; Z.. After perhaps replacing a and b by their negatives we may assume that

ch._. . a and ch._. . b . ra for some r 2 Q, using Proposition 3.13. The elements

_ and b are not uniquely determined but can be varied by adding any integer

multiples of _ and a. The effect of such a variation on the formula ch._. . b . ra

is to change r by an integer, so r is well-defined in the additive group Q=Z, and we

define e.f. to be this element r 2 Q=Z. Since f ' g implies Cf

' Cg , we have a

well-defined map e :_2n−1.S2m.!Q=Z.

Proposition 3.17. e is a homomorphism.

Proof: Let Cf ;g be obtained from S2n by attaching two 2m cells by f and g, so

Cf ;g contains both Cf and Cg . There is a quotient map q :Cf.g!Cf ;g collapsing

a sphere S2m−1 that separates the 2m cell of Cf ;g into a pair of 2m cells. In the

upper row of the commutative diagram at the right we

¡!

¡¡¡¡¡!

¡¡!

K(C )

H (C ; )

e Q ¤

q¤

f, g f g

¡!

K C

( )

H ( ; )

e Q ¤

g f +g

have generators _f and _g mapping to _f.g and _f ;g

mapping to _f.g , and similarly in the second row with

generators af , ag , af.g , bf ;g , and bf.g . By restriction

to the subspaces Cf and Cg of Cf ;g we obtain ch._f ;g. . bf ;g

. rfaf

. rgbg, so

ch._f.g. . bf.g

. .rf

. rg.af.g . tu

There is a commutative diagram involving the double suspension:

S ¡¡S!

e e

¼ ( m) + 2 ¼ (S )

2 2 2m 2

n - 1 2n +1

¡¡¡¡¡!

Stiefel-Whitney and Chern Classes Section 3.2 79

Commutativity follows from the fact that CS2f

. S2Cf and ch commutes with the

double suspension, as we saw in the proof of Proposition 3.9. From the commutativity

of the diagram there is induced a stable e invariant e :_s

2k−1!Q=Z for each k.

Theorem 3.18. If the map f : S2k−1!U.n. represents a generator of _2k−1.U.,

then e.JCf. . __k=k where _k is defined via the power series

x=.ex − 1. .

i _ixi=i!

Hence the image of J in _s

2k−1 has order divisible by the denominator of _k=k.

The numbers _k are known in number theory as Bernoulli numbers. After proving

the theorem we will show how to compute the denominator of _k=k.

Recall from the beginning of x2.4 that the Thom space T.E. of a vector bundle

E!X is defined to be the quotient D.E.=S.E. of the unit disk bundle of E by the unit

sphere bundle. Just as in K–theory, the Thom isomorphism for ordinary cohomology

can be viewed as an isomorphism Ø :H_.X. _ eH

_.T .E.. since the latter group is

isomorphic to H_.D.E.; S.E... Thom spaces arise in the present context through the

following:

Lemma 3.19. CJf is the Thom space of the bundle Ef!S2k determined by the

clutching function f : S2k−1!U.n..

Proof: By definition, Ef is the union of two copies of D2k_Cn with the subspaces

@D2k_Cn identified via .x;v. _ .x; fx.v... Collapsing the second copy of D2k_Cn

to Cn via projection produces the same vector bundle Ef , so Ef can also be obtained

from D2k_CnqCn by the identification .x;v. _ fx.v. for x 2 @D2k . Restricting to

the unit disk bundle D.Ef ., we have D.Ef . expressed as a quotient of D2k_D2n q

D2n

0 by the same identification relation, where the subscript 0 labels this particular

disk fiber of D.Ef .. In the quotient T.Ef . . D.Ef .=S.Ef . we then have the sphere

S2n . D2n

0 =@D2n

0 , and T.Ef . is obtained from this S2n by attaching a cell e2k.2n

with characteristic map the quotient map D2k_D2n!D.Ef .!T.Ef .. The attaching

map of this cell is precisely Jf , since on @D2k_D2n it is given by .x;v.,fx.v. 2

D2n=@D2n and all of D2k_@D2n maps to the point @D2n=@D2n . tu

To compute eJC.f . we need to compute ch._. where _ 2 e K.CJf . . e K.T.Ef ..

restricts to a generator of e K.S2n.. Such a _ is a K theory Thom class since the S2n

here is D2n

0 =@D2n

0 for a fiber D2n

0 of D.Ef .. Recall from Example 2.28 how we constructed

a Thom class U 2 eK

_.T .E.. for a complex vector bundle E!X via the short

exact sequence

0 -!eK

_.T.E..!- K_.P.E_1.. --!-_ K_.P.E.. -!0

with U mapping to

i.−1.i_i.E.Ln−i . A similar construction can also be made with

ordinary cohomology. The defining relation for H_.P.E.. as H_.X. module has

80 Chapter 3 Characteristic Classes

the form

i.−1.ici.E.xn−i . 0 where x . x.E. 2 H2.P.E.. restricts to a generator

of H2.CPn−1. in each fiber. Viewed as an element of H_.P.E_1.., the element

i.−1.ici.E.xn−i , with x . x.E_1. now, generates the kernel of the map to

H_.P.E.. since the coefficient of xn is 1. So

i.−1.ici.E.xn−i 2 H_.P.E_1.. is

the image of a Thom class u 2 H2n.T .E... For future reference we note two facts:

(1) x . c1.L. 2 H_.P.E_1.., since the defining relation for c1.L. is x.L.−c1.L. . 0

and P.L. . P.E_1., the bundle L!E_1 being a line bundle, so x.E_1. .

x.L..

(2) If we identify u with

i.−1.ici.E.xn−i 2 H_.P.E_1.., then xu . 0 since the

defining relation for H_.P.E_1.. is

i.−1.ici.E_1.xn.1−i . 0 and ci.E_1. .

ci.E..

For convenience we shall also identify U with

i.−1.i_i.E.Ln−i 2 K.P.E_1...

We are omitting notation for pullbacks, so in particular we are viewing E as already

pulled back over P.E_1.. By the splitting principle we can pull this bundle E back

further to a sum

i Li of line bundles over a space F.E. and work in the cohomology

and K–theory of F.E.. The Thom class u .

i.−1.ici.E.xn−i then factors as a

product

i.x − xi. where xi

. c1.Li., since ci.E. is the ith elementary symmetric

function _i of x1; ___;xn. Similarly, for the the K–theory Thom class U we have

U .

i.−1.i_i.E.Ln−i . Ln_t.E. . LnQ

i_t.Li. . LnQ

i.1 . Lit. for t . −L−1, so

U .

i.L − Li.. Therefore we have

ch.U. .

ich.L − Li. .

i.ex − exi. . u

i..exi − ex.=.xi

− x..

This last expression can be simplified to u

i..exi − 1.=xi. since after writing it as

iexi

i..1−ex−xi.=.xi

−x.. and expanding the last product out as a multivariable

power series in x and the xi ’s we see that because of the factor u in front and the

relation xu . 0 noted earlier in (2) all the terms containing x can be deleted, or what

amounts to the same thing, we can set x . 0.

Since the Thom isomorphism Ø for cohomology is given by cup product with the

Thom class u, the result of the preceding calculation can be written as Ø−1ch.U. .

i..exi − 1.=xi.. When dealing with products such as this it is often convenient to

take logarithms. There is a power series for log..ey − 1.=y. of the form

j _jyj=j!

since the function .ey − 1.=y has a nonzero value at y . 0. Then we have

log Ø−1ch.U. . log

i..exi − 1.=xi. .

log..exi − 1.=xi. .

i;j _jxj

i =j!

j _jchj.E.

where chj.E. is the component of ch.E. in dimension 2j . Thus we have the general

formula log Ø−1ch.U. .

j _jchj.E. which no longer involves the splitting of the

bundle E!X into the line bundles Li , so by the splitting principle this formula is

valid back in the cohomology of X.

Stiefel-Whitney and Chern Classes Section 3.2 81

Proof of 3.18: Let us specialize the preceding to a bundle Ef!S2k with clutching

function f : S2k−1!U.n. where the earlier dimension m is replaced now by k. As

described earlier, the class _ 2 e K.CJf . . e K.T.Ef .. is the Thom class U , up to a sign

which we can make .1 by rechoosing _ if necessary. Since ch.U. . ch._. . b.ra,

we have Ø−1ch.U. . 1 . rh where h is a generator of H2k.S2k.. It follows that

log Ø−1ch.U. . rh since log.1.z. . z −z2=2.___ and h2 . 0. On the other hand,

the general formula log Ø−1ch.U. .

j _jchj.E. specializes to log Ø−1ch.U. .

_kchk.Ef . in the present case since eH

2j.S2k;Q. . 0 for j . k. If f represents a

suitable choice of generator of _2k−1.U.n.. then chk.Ef . . h by Proposition 3.13.

Comparing the two calculations of log Ø−1ch.U., we obtain r . _k . Since e.JCf.

was defined to be r , we conclude that e.JCf. . _k for f representing a generator of

_2k−1.U.n...

To relate _k to Bernoulli numbers _k we differentiate both sides of the equation

k _kxk=k! . log..ex − 1.=x. . log.ex − 1. − log x, obtaining

k_1 _kxk−1=.k − 1.! . ex=.ex − 1. − x−1 . 1 . .ex − 1.−1 − x−1

. 1 − x−1 .

k_0 _kxk−1=k!

. 1 .

k_1 _kxk−1=k!

where the last equality uses the fact that _0 . 1, which comes from the formula

x=.ex − 1. .

i _ixi=i! . Thus we obtain _k

. _k=k for k > 1 and 1 . _1 . _1. It is

not hard to compute that _1 . −1=2, so _1 . 1=2 and _k

. −_k=k when k . 1. tu

The numbers _k are zero for odd k > 1 since the function x=.ex −1.−1.x=2 .

i_2 _ixi=i! is even, as a routine calculation shows. Determining the denominator of

_k=k for even k is our next goal since this tells us the order of the image of eJC in

these cases.

Theorem 3.20. For even k > 0 the denominator of _k=k is the product of the prime

powers p`.1 such that p −1 divides k and p` is the highest power of p dividing k.

More precisely:

(1) The denominator of _k is the product of all the distinct primes p such that p−1

divides k.

(2) A prime divides the denominator of _k=k iff it divides the denominator of _k .

The first step in proving the theorem is to relate Bernoulli numbers to the numbers

Sk.n. . 1k . 2k . ___ . .n − 1.k .

Proposition 3.22. Sk.n. .

Pk i.0

_k−ini.1=.i . 1..

Proof: The function f.t. . 1.et .e2t .___.e.n−1.t has the power series expansion

Xn−1

`.0

k.0 `ktk=k! .

k.0 Sk.n.tk=k!

82 Chapter 3 Characteristic Classes

On the other hand, f.t. can be expressed as the product of .ent−1.=t and t=.et−1.,

with power series

i.1 niti−1=i!

j.0 _jtj=j! .

i.0 ni.1ti=.i . 1.!

j.0 _jtj=j!

Equating the coefficients of tk we get

Sk.n.=k! .

i.0 ni.1_k−i=.i . 1.!.k − i.!

Multiplying both sides of this equation by k! gives the result. tu

Proof of 3.20: We will be interested in the formula for Sk.n. when n is a prime p:

._. Sk.p. . _kp .

_k−1p2=2 . ___._0pk.1=.k . 1.

Let Z.p.

_ Q be the ring of p integers, that is, rational numbers whose denominators

are relatively prime to p. We will first apply ._. to prove that p_k is a p integer

for all primes p. This is equivalent to saying that the denominator of _k contains no

square factors. By induction on k, we may assume p_k−i is a p integer for i > 0.

Also, pi=.i . 1. is a p integer since pi _ i . 1 by induction on i. So the product

_k−ipi.1=.i . 1. is a p integer for i > 0. Thus every term except _kp in ._. is a

p integer, and hence _kp is a p integer as well.

Next we show that for even k, p_k

_ Sk.p. mod p in Z.p. , that is, the difference

p_k

− Sk.p. is p times a p integer. This will also follow from ._. once we see that

each term after _kp is p times a p integer. For i > 1, pi−1=.i . 1. is a p integer by

induction on i as in the preceding paragraph. Since we know _k−ip is a p integer, it

follows that each term in ._. containing a _k−i with i > 1 is p times a p integer. As

for the term containing _k−1 , this is zero if k is even and greater than 2. For k . 2,

this term is 2.−1=2.p2=2 . −p2=2, which is p times a p integer.

Now we assert that Sk.p. _ −1 mod p if p−1 divides k, while Sk.p. _ 0 mod p

in the opposite case. In the first case we have

Sk.p. . 1k . ___ . .p − 1.k _ 1 . ___.1 . p −1 _ −1 mod p

since the multiplicative group Z_

. Zp

−f0g has order p−1 and p−1 divides k. For

the second case we use the elementary fact that Z_

p is a cyclic group. (If it were not

cyclic, there would exist an exponent n < p−1 such that the equation xn −1 would

have p−1 roots in Zp , but a polynomial with coefficients in a field cannot have more

roots than its degree.) Let g be a generator of Z_

p , so f1; g1; g2; ___ ; gp−2g . Z_

p. Then

Sk.p. . 1k . ___ . .p − 1.k . 1k . gk . g2k . ___.g.p−2.k

and hence .gk − 1.Sk.p. . g.p−1.k − 1 . 0 since gp−1 . 1. If p − 1 does not divide

k then gk . 1, so we must have Sk.p. _ 0 mod p.

Statement (1) of the theorem now follows since if p − 1 does not divide k then

p_k

_ Sk.p. _ 0 mod p so _k is p integral, while if p −1 does divide k then p_k

Sk.p. _ −1 mod p so _k is not p integral and p divides the denominator of _k .

To prove statement (2) of the theorem we will use the following fact:

Stiefel-Whitney and Chern Classes Section 3.2 83

Lemma 3.23. For all n 2 Z, nk.nk − 1._k=k is an integer.

Proof: Recall the function f.t. . .ent − 1.=.et − 1. considered earlier. This has

logarithmic derivative

f 0.t.=f .t. . .log f.t..0 . .log.ent − 1. − log.et − 1..0 . nent=.ent − 1. − et=.et − 1.

We have

ex=.ex − 1. . 1=.1 − e−x. . x−1.−x=.e−x − 1.. .

k.0.−1.k_kxk−1=k!

f 0.t.=f .t. .

k.1.−1.k.nk − 1._ktk−1=k!

where the summation starts with k . 1 since the k . 0 term is zero. The .k − 1.st

derivative of this power series at 0 is _.nk−1._k=k. On the other hand, the .k−1.st

derivative of f 0.t..f .t..−1 is .f .t..−k times a polynomial in f.t. and its derivatives,

with integer coefficients, as one can readily see by induction on k. From the formula

f.t. .

k³0 Sk.n.tk=k! derived earlier, we have f .i..0. . Si.n., an integer. So the

.k − 1.st derivative of f 0.t.=f .t. at 0 has the form m=f .0.k . m=nk for some

m 2 Z. Thus .nk − 1._k=k . _m=nk and so nk.nk − 1._k=k is an integer. tu

Statement (2) of the theorem can now be proved. If p divides the denominator

of _k then obviously p divides the denominator of _k=k. Conversely, if p does not

divide the denominator of _k , then by statement (1), p − 1 does not divide k. Let g

be a generator of Z_

p as before, so gk is not congruent to 1 mod p. Then p does not

divide gk.gk − 1., hence _k=k is the integer gk.gk − 1._k=k divided by the number

gk.gk − 1. which is relatively prime to p, so p does not divide the denominator of

_k=k.

The first statement of the theorem follows immediately from (1) and (2). tu

There is an alternative definition of e purely in terms of K–theory and the operations

k . by the argument in the proof of Theorem 2.17 there are formulas k._. .

km_ and k._. . kn_._k_ for some _k

2 Z satisfying _k=.km−kn. . _`=.`m−`n..

The rational number _k=.km − kn. is therefore independent of k. It is easy to check

that replacing _ by _ . p_ for p 2 Z adds p to _k=.km − kn., so _k=.km − kn. is

well-defined in Q=Z.

Proposition 3.24. e.f. . _k=.km − kn. in Q=Z.

Proof: This follows by computing ch k._. in two ways. First, from the formula

for k._. we have ch k._. . knch._. . _kch._. . knb . .knr . _k.a. On the

other hand, there is a general formula chq k._. . kqchq._. where chq denotes the

component of ch in H2q . To prove this formula it suffices by the splitting principle

and additivity to take _ to be a line bundle, so k._. . _k , hence

chq k._. . chq._k. . .c1._k..q=q! . .kc1._..q=q! . kqc1._.q=q! . kqchq._.

84 Chapter 3 Characteristic Classes

In the case at hand this says chm k._. . kmchm._. . kmra. Comparing this with

the coefficient of a in the first formula for ch k._. gives _k

. r.km −kn.. tu

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