Delooping the Quillen map.

DELOOPING THE QUILLEN MAP

by

Jrgen Tornehave Cand. Scient. Aarhus

(1967)

University

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF

DOCTOR OF PHILOSOPHY at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1971

Signature of Author,. Signature redacted

Department of Mahematics, June, 1971

Signature redacted

Certified by.. ... ,... .h .... Accepted by....

Signature redacted

Chairman,

Archives

JUN 15 1971

LIBR A RO E Departmental Committee on Graduate Students

(V&{

DELOOPING THE QUILLEN MAP by

Jdrgen Tornehave

Submitted to the Department of Mathematics on May 14, 1971 in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

ABSTRACT

By means of a character computation it is shown that the natural transformation of homotopy functors, induced by the Quillen map for a finite field F,

BF -+ BU, commutes with the trace homomorphisms for

finite coverings. Moreover it is shown that this state-ment does not hold for the map BF x Z -+ BU x Z.

Next it is shown that the ma obtained by

localization BF X _F Z + BU[-] _p x Z[-], where p = char F, has a unique multiplicative infinite delooping. From this is deduced that BF + BU is an infinite loop map.

Thesis Supervisor: D. W. Anderson

Title: Associate Professor of Mathematics

3

.

TABLE OF CONTENTS

0 Introduction ₄

1 Wreath products of- representations 9

2 Commutativity with the trace 21

3 The K-theories associated to finite fields 36

4 _{Proof of theorem 3.6 44}

5 Extension to infinite CW-complexes 74

Bibliography ₈₆

4.

0. INTRODUCTION

There has been some recent interest in ways of constructing infinite loop spaces from certain kinds of functors on finite basepointed sets, i.e. the r-spaces of G. Segal [16] or the chain functors of D. Anderson [4],

[5]. Moreover Anderson has defined a functor from

permutative categories (Or even simplicial permutative categories) to chain functors, and he has shown that

permutative categories exist in abundance. In particular to any field F there is associated a permutative

category, which gives rise to an infinite loop space structure on

QBiLBGL(n, F)

and hence to a generalized cohomology theory k

This construction generalizes to simplicial rings, and when applied to the singular complex of C, the field

of complex numbers, it gives rise to connected complex K-theory kC. _{The coefficients of the theory kF in} negative degrees are exactly the algebraic K-groups of F as defined by Quillen [15]. Quillen has computed

the coefficients in field.

Now let F be closure of a finite

the case, where F is a finite

a finite field or the algebraic field. We have

QBiLBGL(n, F) = BF x Z

where BF is the zero component of the left hand space. In his proof of the Adams conjecture [14], Quillen uses a map

BGL(co, F) -+ BU,

which is constructed by means of Brauer's theory of modular characters. This map extends to

f : BF + BU,

which we refer to as the Quillen map.

The main purpose of this work is to prove that the Quillen map is an infinite loop map. Moreover we prove that

U.

* x 1 : BF x Z + BU x Z

is not an infinite loop map, but that it becomes an infinite loop map, when we localize away from the prime p = char F in the sense of Sullivan [17]

(that is, we invert the single prime p).

In 1 and 2 of this work we prove that the Quillen map

*

commutes with the trace for finite coverings. This appears to be saying that the first order obstructions of a whole series of higher order obstructions to delooping a map, vanish. The author hopes to clarify this in the future. The purely

representation theoretic part of this is contained in 1, where we define wreath products of representations, and prove a result about the behavior of wreath products under modular lifting (theorem 1.11). In 52 we translate this result into the commutativity of $ with the trace, and prove that x 1 : BF x Z + BU x Z does not commute with trace.

In 3, 4 and 5 we attack the problem from a different angle using much more information about the

cohomology theories k , in particular the existence of a product (Anderson [4], [5]). We recall the results

I( ₀

of Quillen [15] in 3 and state theorem 3.6 about the existence and uniqueness of a stable multiplicative operation

k;(

)

+1C

6Z

kk(

)

Z[1]

F C p

defined on the category of finite CW-cotnpleies, and agreeing with the Quillen map in degree 0.

The proof of theorem 3.6 is contained in 4. The first step is to use the technique of 1 and 2'in a much simpler situation to prove that

*

commutes with the product maps in degree 0. The second step is to use periodicity to construct operations

k *(-) +b K* (-; Z/1)

where X is prime to p. To do this we need some of the results from Araki and Toda [6] about products in

* *

kC(-; Z/1) and kF(-; Z/t). The third step uses a technique from Anderson [3] to lift the operation from

the kC(-; Z/)'s to kC(-) 0 Z[11.

C C p

In 5 we represent

*

as a multiplicative map of spectra by means of the "lim -technique", thus proving

-4-8.

that

B x Z + BU ] x Z[-]

Fp p

is an infinite loop map. It follows easily that the Quillen map is an infinite loop map.

9.

1. Wreath Products of Representations

Let 0 be a commutative ring with unit and G a finite group. A representation of G over 0 is defined to be a projective finitely generated n-module

E with a P-linear action of G on E. The representation ring of R (G) of G over 0 is the free abelian group generated by the isomorphisms classes of representations of G over n, modulo the relations E = El + E"

arising from short exact sequences of representations

(1 ) 0 +El -+ E + E"l+ 0

A product in RQ(G) is induced by tensor product of representations.

The ring R (G) is a contravariant functor in G and a covariant functor in 0. A ring homomorphism o -. r induces the homomorphism

RQ(G) + Rr(G)

sending E into E

e

Q r

We will mostly be concerned with case, where 0 is the field of complex numbers C or the algebraic

10.

closure 7 of a finite field. We shall also use the homomorphism

V

(1.2) R (G) RC(G)

given by modular character theory. Let

(1.3) 1 : T* + C

be an embedding of the multiplicative group of

F

into the roots of unity in the multiplicative group C . For a representation E of G over

F

we define the

modular character XE of E to be the function G - C given by

(1.4)

E(g)

where the summation is over the eigenvalues of g : E -+ E counted with multiplicities.

We shall use another description of (1.2).

Let K be a finite field extension of Q, which is a splitting field of G. Then

11.

RK(G) -+ R C(G)

is an isomorphism. _{Further let 0 be the ring of}

algebraic integers in K, and choose P E Spec 9 above p = char 7 E Spec Z. The localization n is a

discrete valuation ring with K as the quotient field and a finite residue class field F = aP/Pop of

characteristic p. Finally choose an embedding j : F + T. It is a basit fact from the theory of

modular representations (Curtis and Reiner [9], thm. 82.1) th&t the inclusion f2 + K induces an &pimorphism

RQ (G) + RK(G),

and that there exists a (consequently unique) homomorphism

d : RC G) + Ry(G),

called the decomposition homomorphism, such that the following diagram commutes

R (G) > RC(G)

(1.5) 1

Next we observe that RC(G) and Rr(G) are X-rings. _{It is easily seen from (1.5), that d is a} X-ring homomorphism. _{In particular d commutes with} the, Adams operations * .

Now let e = e(G) be write _{e = s opa, where} s power of p, q = p b, with From the character formula

the exponent of is prime to p. b > a and q = for E c RC(G), G, and Choose a 1 (mod s). (1.6) _{Xi () = XE(g i)}

is seen that *q is idempotent on RC(G). Another fact, we shall need, is that the embedding j : F + 7 can

be chosen in a unique way so as to make the following diagram commutative (Quillen [14] 5.4.2)

IpR (G)

FC (G)

Rry(G) ( . ) ( J

In particular v map Ryr(G) isomorphically onto VRC(G)o It follows that (1.2) commutes with the Adams operations, and is a ring homomorphism.

We let En denote the symmetric group on {l, 2,..., n} and recall that the wreath product En f G for any group G is the semidirect product

En f G = En x (Gx...xG) (n G's)

where En acts on Gx...xG by permutation of

coordifates. Explicitly the product in E n G is

given by

(1.8) (a, g₁,..., gn)(T, h1, .. , hn)

agga _{.. gna-}

(n)

For a representation E of G over a ring 0 we define the wreath product En f E as the representation

of En I G in the a-module EOn given by

(1.9) _{(a, gl,..,,gn (el****}

= (gle _{,...,gne -1(n)}

L4

n-fold exterior direct sum of E and En n f G

acts on E n by permutation.

Clearly a short exact sequence (1.1) induces a short exact sequence

0 -+ En fE' + Eb f E -+ n f E" 0

of En f G-modules. Cdnsequently for any ring 0 we can define an additive homomorphism

(1.10) En : Rg(G) + a%(En f G)

by Ew+ InF. E.

In the following theorem we use the notation RC(G) for the I(G)-adic completion of RC G), where

the ideal I(G) is the kernel of the dimension

homomorphism RC (G) + Z. Similarly let Igr(G) be the kernel of dim : Ry(G) + Z.

Theorem 1.11 For every finite group G the following diagram is commutative

Igr(G) R C(G) En n 15. Ry(En f G) RC(E n f G)

We give the proof after some preparation. Lemma 1.12. If Q - r is a ring homomorphism,

the following diagram is commutative

R (G) n > R f(E G)

Rr(G) r ( n G)

Proof. This ia immediate from the definitions. Lemma 1.13. The decomposition homomorphism commutes with wreath product, i.e. the diagram

RC(G) n f> RC (n G)

I d

Id

16.

commutes.

Proof. _{Let K be a finite field extension of Q,}

which is a splitting field for both G and E f G.

,n

Choose P c Spec Q and the embedding j : F + F that makes the diagrams (1.7) for G and En IG _dommutative

(this choice depends only on the previously chosen

* *

embedding 7 + C ). Since the homomorphism

RSI (G) +RC (G)

P

is surjective, the lemma follows from (1.5) applied to G

and En f G combined with lemma 1.12.

This lemma reduces the theorem to a problem about the compatibility of En f with the particular Adams operation *q, that occurs in (1.7). For this we need the following character formula.

Lemma 1.14. For E e R C(G) the character of

En E is given by

XE

I

E(a, g1 se*'''n) =

Z

XE i

n a(i)=i

1.70

summation is over the set of fixed points under the action of a on {l,...,n}.

P"oof. By linearity it suffices to prove this for an actual representation E. Let g : E -+ E have the matrix _{(ast(g)) with respect to a basis}

e ,...,er of E, and consider the matrix of

(a, g,...,gn ) : E On +o E On

with respect to the basis "e1,...,er repeated n times". This matrix decomposes into rxr - blocks each being 0 or of the form (a ₅ (gi)). In particular the i'th

diagonal block is 0, if a(i) # i, and (a st(g)), if a(i) = i. It follows that the trace of (a, g,...,gn on E n is the sum of the lemma.

Finally we shall use the following result of Atiyah ([7] 699).

Lemma 1.15. The element E E R (G) belongs to the kernel of

18.

if and only if XE(g) = 0 for every element g e G of prime power order.

Proof of theorem 1.11. Let e be the exponent of Zn I G and write e = s *pa with S prime to p. Pick b > a such that q = pb 1 (mod s). This choice makes the diagrams (1.7) for both G and En f G

commute. Consider an element E' c Ig(G) and let E = vE c RC(G).. Then by lemma 1.13 and (1.7)

V(En f E') = v(En f dE) = vd(En f E) = $p9(En f E)

The character XV(E (1.8) and lemma 1.14.

E') can be computed using (1.6),

(1.16) XV(En I El)(a 2 ***n) = XEn f E((a *** n

= XE

_n

'

E

(

*** i F1 U) n a (i) -q XE gi -1 * _-q+U * -q+ " * * )

)

By lemma 1.15 we only have to prove that this -sum is equal to

(1.17) XE

f

E(o, gl**gn) "XE

1-n a(i)-i

whenever (a, g ,...,gn) e En I G is of prime power order. Since q E = E,

XE81)

XE(gq),

and the sum (1.17) is contained in the sum (1.16).

if (a, g ,,n) is of order prime to p, so is a e En. But then a and a have the same fixed poitts, and the two sums are equal. If (a, gl,...,gn is of order a power of p, we get

1 = (G~gg,...,gn) = (a, g ... g a-q+l ()***)

and since dim E = dim E' = 0 all the terms in (1.16) and (1.17) vanish. This proves the theorem.

Remark. Theorem 1.11 with I7(G) replaced by RF(G) does not hold if n > p = char 7. To see this we only have to look at the unit 1 e R ₁(G). The

formulas of the above proof show that

Xv(En f 1) (a El,****9n) = no. of fixed points of aq

These numbers are different when a is a cycle of length p.

21.

2. Commutativity with the Trace

To every finite field F = F (the field with q elements) there is associated a 0-connected Q-spectrum BF with the space

(2.1) _{B F = (QB.L[BGL(m, F))} 0 m=0

in degree 0. Here the disjoint union IIBGL(m, F) has a topological monoid structure induced by direct sum

GL(ml, F) x GL(m₂, F) -+ GL(m₁ + m₂ , F),

and ( )0 means zero component. The inclusions of the suspensions

SBGL(m, F) + BI.BGL(m, F)

give homotopy classes

BGL(m, F) + BF,

BGL(m, F) + BF

The theorem of Barratt and Priddy [8] implies that

(2.2) induces an isomorphism on integral cohomology and hence also on complex K (see lemma 2.12).

For each m we have the element

(2.3) Em e RF(GL(m, F))

defined to be the standard representation on F m minus the trivial m-dimensional representation, The modular liftings

vEm c RC(GL(m, F))

determine elements in

K(BGL(m, F)) = RC(GL(m, C)),

that can be combined to an element in

K(BF) = K(BGL(m., F)) = lim K(BGL(m, F)).

Here the last isomorphism follows from the exact

23. sequence of Milnor [13]

0 lim 1 _K1_(BGL(m,F))

-+ K(BGL(w,F)) +1 lim K(BGL(m,F)) + 0

.4-

4-and the fact that K (BG) = 0 for every finite group G (Atiyah [7]). _{Hence we have a map}

* : BF + BU

well defined up to homotopy modulo the choice of an embedding of the multiplicative group F into the group of roots of unity in C.

Since BF and BU are the "bottom" spaces of 0-connected 0-spectra, there are well defined trace homomorphisms

[7, BF] [X, BF]

(2.5)

.Ir

[X, BU2 -> [X, BF1

where w :

X

+ X is a finite basepointed covering. Theorem 2.6. The homomorphism

B-

BF --- > [-, BU]

commutes with trace. Hence for every finite basepointed covering w : I + X we have a commutative diagram

[X, BF] -> [, BU]

[C, B] -> [X, BU].

To prove this we shall use the definition of the trace in terms of the Dyer-Lashof maps

EE Bn n +B

n E n F F-9

(2.7)

EEn BUn BU.

An n-fold covering w : - X can be thought of

as a fiber bundle with structure group En* if

r : X -+ X is the associated principal _Zn-bundle,

we can identify % with X/En-1, where Zn-1 is the isotropy group of n for the usual action of En on {l,...,n}. The classifying map X + BEn is covered by

L..J *

a En-equivaraint map

(2.8) _{X o EEn}

A map f : + B. gives rise to a unique Sn-equivariant map

(2.9) X 4 B,

whose composition with the n'th projection pr : BF + BF is equal to the composition

~ f

X + _{-I+ B F'}

We can now combine (2.8) and (2.9) to a En-equivariant map

X n _n

By factoring out by the E n-action we get

X+EEn _E BnF

26.

The composition of this with the Dyer-Lashof map (2.7) represents 1*L[f] e [X, BFJ*

The trace for BU can of course be defined in a similar way.

Remark. _{This construction actually gives a one to} one correspondence between H0-structures on a space B in the sense of Dyer and Lashof [10] and traces on the homotopy functor [-, B]. In this correspondence the homotopy commutativity of the square in [10] involving

the wreath product Zk fn corresponds to the functoriality of the trace.

It is clear from the construction above that theorem 2.6 is a consequence of:

Proposition 2.10. The diagram

EE x Bn n >E x B

EEn X E BFE E E BT n

n I

BF

is homotopy commutative.

27. The maps _{BGL(m, F) + B F} B(E f GL(m, F)) = EZn x induce maps E BGL(m, F)n + Er E Bn and a homomorphism

(2.11) K (En x B ) + lim K *(BE GL(m, F))

n En +

-m

Lemma 2.12. The homomorphism (2.11) is an isomorphism. Hence K (EE x B n) n EnF = 0 K0 (EEn _n E B _En n) A'1 lim R C(En f GL(m, F)) 4-m

Proof. We start from the fact that (2.2) induces

an isomorphism on cohomology with any field as coefficients. Consider the map of coverings

28.

EZn x BGL(, F) E Z n XB

E Z n x BGL(c F + E n x n BF

and the induced map of spectral sequences, which on the E₂-term is

(2.13) H* (BEn; H*(Bn)) + * (BEn; H*(BGL(., F)n)),

where we have omitted the coefficient field. Here we have twisted coefficients given by permutation of

coordinates. It follows that (2.13) is an isomorphism and hence that the map

(-2.14) EE x BGL(o, F)n + EEn Bn

n E n EnF

induces an isomorphism on cohomology. Then the cofiber of this map is acyclic, so that K* of the cofiber is trivial, and

*

C2.14) induces an isomorphism on K *. Now

EEn x E BGL(,, F)n = BEn ! GL(oo, F) = lim BEn f GL(m, F) nm

and

K 1(BEn GL(m, F)) = 0.

Milnor's lemma [13] yields an isomorphism

K(EEx E B n) = lim K(Bn GL(m, F)).

n +

m

Similarly we obtain the isomorphism

K (En x B n)

nEn F lim K(BEn 4- I GL(m, F))

m

For every finite group kernel of

G K(BG) is isomorphic to the

dim : RC(G) _{+ Z,}

which is known to be a profinite group (Atiyah [7]). Moreover the homomorphisms of the inverse system are continuous and hence the lim - term above vanishes, which completes the proof of the lemma.

Proof of proposition 2.10. By lemma 2.12 it is sufficient to prove, that the two compositions

)U.

(2.15) BEn

I

GL(m, C) + En X B F -+ BF BU

nZ

.(2.16) BEn GL(m, C) +) EEn xn B n +,, EEn x BU n + BU

represent the same element in RC(Zn f GL(m, C)). From the construction in Anderson [4] of the

Dyer Lashof maps (they even exist on the "category level") we get a homotopy commutative diagram

BEn GL(m, F) -> EE x B n

n n ZnF

BGL(nm, F) _BF

where the lefthand map is induced by the wreath product of the standard representation of GL(m, F) on Fem By naturality of modular lifting it follows that (2.15)

represents

v(En Em) R₀(En f GL(m, C))

31.

of BGL(m, C), and BI the k-skeleton of

B(En f GL(m, C)). We get a commutative triangle

K(BEn / GL(m, C)

1rM K(EE x E Bn

- _{n n}

lim K(BI) k

Here the bottom map is an isomorphism (Anderson [2]) and the upper map an epimorphism. This shows that the upper map is an isomorphism, so it suffices to consider the composition (2.16) restricted to EEn x B .

n

The composition

B + BGL(m, C) + BF + BU,

which represents the restriction of

32.

to B can be deformed into a map

g : B BU(h)

fpr some h.' The resttiction to En x B n is homotopic n _{n L} of to

lx gn

EE A B nEE x n E6I 1

which agrees with the restriction follows that (2.16) represents

the composition (2.16) the composed map

BU(h)n + BU(nh) + BU,

of En f vEM. It

En f vE e R(C n f GL(m, C))

Proposition 2.10 follows now from theorem 1.11.

We conclude this section by considering the map

(2.17) BF x Z > BU x Z

where the lefthand space is the infinite loop space arising from the permutative category associated to F as in [4], and the righthand space is the infinite loop space arising from the simplicial permutative category

33. associated to C.

Proposition '218, The natural transformation of homotopy functors induced by (2.17) does not commute with the trace for n-fold coverings, when

n > p = char F.

An immediate consequence is:

Corollary 2.19. The map (2.17) is not an infinite loop map.

Proof of prop. 2418. The universal n-fold covering is

BEn-1 _{+ BEn}

where we think of BEn-l as being EE n- , Consider the unit 1 e [BEn-l, BF x Z] represented by the constant map into (*, 1) e BF x Z. From the definition of the

trace in terms of Dyer-Lashof maps is seen, that fr*(l) c [BEn, BF x Z] is given by the composition

(2.20) BEn + EEn x E (BF x Z)n + BF x Z

where the first map is obtained from the En-equivariant map

34.

EEn + EEn $ (BF x Z)n

whose first coordinate is the identity and second map the constant (1, ... , 1) E Z.n We can identify (2.20) with the composition

BEn + BEn f GL(m, F) + BF x Z

vihere the last map goes into B F x {n}. It follows that

x 1)*w*(l) c K(BEn) is represented by the modular lifting of the standard n-dimensional permutation representation of En in F n

Similarly wi*( x 1)*(l) =w*(l) E K(BE n)

is represented by the permutation representation of

Eh in COn The remark at the end of 1 with a being the trivial group shows that these two elements of

K(BE) R (Z) are different, when n > p.

Remark. _{Proposition 2.10, and hence theorem 2.6,} can easily be generalized to the case where F is the algebraic closure of a finite field, since lemma 2.12 and Milnor's lemma give an isomorphism

35.

K(En x Bn) = lim K(EZn x B Bn

n +n n

where the inverse limit is taken over the finite

subfields F' of F (we can pass to a cofinal sequence of finite subfields in order to apply Milnor's lemma).

Clearly proposition 2.18 and its corollary hold for the algebraic closure of a finite field.

3b.

3. The K-theories Associated to Finite Fields

To any field F there is associated a -1-connected

11-spectrum Q(F):

(3.1) Q(O, F), Q(1, F), Q(2, F),

where

Q(O, F) = BF x Z.

The 0-connected covering of (3.1) is the .i-spectrum F mentioned at the beginning of 2.

The spectrum (3.1) can be constructed either from a r-space (Segal [16]) or a permutative category

(Anderson [4], [5]).

Anderson uses the tensor product over F to define a "semiring structure" on the permutative category associated to F. This in turn induces a uSteenrod structure" on Q(F). In particular we have product maps

Q(i, F) A Q(J, F) + Q(i + j, F), (3.2)

-) 1

which on the associated cohomology theory k define a commutative and associative product with unit.

In the case F = C we let (3.1) denote the spectrum for connected complex K-theory kC. This

can actually be obtained from the simplicial permutative category associated to the singular complex of C

considered as a simplicial ring (Anderson E5]).

A field extension F + F' induces a map of spectra

Q(i, F) + Q(J, F'),

which homotopy commutes with (3.2). Hence we have a stable multiplicative cohomology operation

k * k ,t

Let F = Fq be the finite field with q = pa (p a prime) elements and 7 q

7 its algebraic closure.

We have the Quillen map

BF BU

that the composition

BGL(m, F q) + B + BU

is represented by the element in R (GL(m, F ))

arising by modular lifting of the reduced standard representation of GL(m, F ) on F m

q q

An extension of finite fields Fq + Fq gives rise to a homotopy commutative diagram

BF

q

BU

Fq1

Since K (BF ) = 0 (lemma 2.12 with n = 1) we get from Milnor's lemma an isomorphism

3j :;

where the inverse limit is taken ever the category of finite subfields of !. Thus we have a map uniquely

determined up-to homotopy

Bt + BU

This is the Quillen map for the algebraic closure of a finite field.

Consider the following diagram

MI Eqp

looI

B F BU

q

BU

where E$ q is the fiber of *q - 1. Quillen shows in

[15], that ( - )$ is homotopic to a constant, so that a lifting BF ++ E- q of

*

exists, and tbat

q

this lifting is a homotopy equitalenoe. Note that,

40.

An examination of the exact homotopy sequence for the fibration then- gives Quillen's result on the algebraic K-groups of finite fields:

k (21-l)(pt) = w2i-(BF ) Z

q q q -1

(3.3) (i > 0)

k-21(pt) = h2 (BF ) = 0

q q

It is easy to determine the coefficient ring k (pt) from this.

Lemma 3,4. The inclusion F - F s induces

multiplication by

s-1 is

E q d = q

J=0 q -l

r21.-l(BF q 21-1 BF

Proof. We have the following homotopy commutative diagram

41.

U U -> -> BU > BU

U >U E* s---> BU 94~>BU

From the corresponding homomorphism of exact homotopy

k

sequences and the fact that

'P

: U + U induces

multiplication by ki on 721-l, it is seen that the homomorphism

w21-1 (E* q) +, 721- (E9qs) s-1i

is multiplication by E q . Using the fact that

J=o

K (BF ) - 0 we see that the diagram

q

BF _{-- > BF}

q qs

EI.

-->

'

4 2.

Proposition 3.5. The coefficients of kp are given by

-k (2 - l)(pt) =

- 2j-1(Br) = Z(p)/Z

(i > 0)

k 21(pt) = W2i(Bw) 0

where p = char F, and Z(p) is the localization of the integers at the prime ideal (p) generated by p.

Proof. We have

Wi(By) = lim w(B,,)

where the direct limit is taken over the category of finite subfields of 7. The result follows easily from (3.3) and lemma 3.4.

Remark. The product structure in k.(pt)

must be trivial for dimension reasons. More precisely any two homogeneous elements of negative degree in kp(pt) has product zero, and we have a unit

0

43.

The map

Z' BU x Z ,

where F is a finite field or the algebraic closure of a finite field, induces a natural homomorphism

k0 k0

kF( - ) +kC

The next section is devoted to the proof of the following theorem.

Theorem 3.6. Let F be a finite field or the

algebraic closure of a finite field of characteristic p. Then the composition

k

k - 0 k1 Z[

has a unique extension to a multiplicative stable cohomology operation

S: k -

)

+k - )

0

Z[I]

d hp

defined on the category of finite CW-complexes.

I

4. Proof of Theorem 3.6

Let F be the algebraic closure of a finite field of characteristic p. We form the diagram

BF A BF BU A BU

(4.1)

'p

'Nit

BF >BU

where the vertical maps are obtained by restricting the product maps

Q(0, F) A

Q(o,

F)

Q(O, C) A Q(O, C)

+ Q(0, F)

+ Q(0, C)

to the zero components.

Proposition 4.2. The diagram (4.1) is homotopy commutative.

Proof. Because of the exact sequence

45.

it suffices to prove that the two compositions

B F x B F BF -> BU

BF x BF - -> BU x BU >BU

are homotopic, Moreover the isomorphism (compare lemma 2.12)

K(BF x BF) = _{lim lim K(BGL(n, F') x BGL(m, F')),}

.4--F' n,m

where F1 runs over the finite subfields of F, reduces it to proving, that the two compositions

BGL(n, FI) x BGL(m, F') + B , x BF, + BF, + BU BGL(n, FI) x BGL(m, F') + BF, x BF, + BU x BU -+ BU

are homotopic.

Let En and Em be the reduced standard representations of GL(n, F') and GL(m, F') respectively. Then the

46. of the image of

En 0 Em e RF(GL(n, F')) 0 RF(GL(m, F'))

in

RF(GL(n, F') x GL(m, F'))

under the product map. The second composition is represented by the product of the modular liftings of En and E m Hence it suffices to prove commutativity

of the diagram

RF(G1) 0 RF(G₂) - RF(Gl x G₂)

i

v V v

RC(G₁ ) 0 RC(G₂) + RC (G x G₂)

for two finite groups G1 , 020 This in turn follows

from the fact that modular lifting

L+7.

is a ring homomorphism for every finite group G. Corollary 4.3. The homomorphism

k0 k0 K

$ : kF( - ) k( ) - )

commutes with product.

Proof. Proposition 4.2 shows that this holds for the corresponding homomorphism of reduced groups

k F(X) _kF(X)

for any choice of a basepoint in a connected complex X. The corollary follows since $ maps 1 e k0(X) into 1 C k (X).

C

In the rest of this section we will only work with reduced cohomology theories defined on the category of finite CW-complexes with basepoint.

For k > 2 a natural number let

M = U 2

standard degree X map. For a generalized cohomology theory h one can define cohomology theories

h (-; Z/L) with

hi(-; Z/1) = h1+2(- MX).

We will use Araki and Toda [6] as our source of

information on these matters (see also Anderson [3] and Maunder [12]).

We define the natural transformation

(4.4) $ : -2 (-; Z/A) k+ ~2(-; Z/L)

to be

0 ~10

* ; k (- M) + kC(- Mz).

Proposition 4,5. For I prime to p the natural transformation (4.4) is an isomorphismi

Proof. Since both sides of (4.4) are homotopy functors it suffices to get an isomorphism on spheres, that is to prove that

L~ 1)

$ : kS(SiML) + k0(S M ) = K(S) M

is an isomorphism for every i > 0. From the exact sequence

0 21+1) F

21+) + 21-1 ) 21

kF(S kF 3.M k (Sn kF(Si

and proposition

3,5

ffoll~ows

kO(S2 ilM ) = 0 _{(i > 1).}

The exact sequence

K( _o 21+l) + K(s21-1M ) _{(21) z K(21} SKK

Kt(hS

shows that

K(s21-lM ) = 0 (U 1)

Next the exact sequence

50.

and proposition 3.5 yields

kF (S2M L

Finally from the

Z/X (i > 0). exact sequence (4.6) K(321+2) K(S21+2

)

K(S21M) K K(S21+) we get K(S21M ) = Z/. (i > 0 ).

To prove that

*

induces an isomorphism it

suffices to find a finite subfield Fq of F, such that

$ : k2 (s2iM) +(s21Mz)

is an epimorphism. Using the homotopy equivalence BF + Etp we get an exact sequence

Hence it suffices to find a power q = qa of p,

such that *q. acts as the identity on K(S21M). From

(4.6) and the fact that *q acts on K(S21+2) as

multiplication by qi+l, we see that q only has to

be chosen, so that

qi+l E 1 (mod X),

This is possible, when p is prime to L.

Let h be a generalized cohomology theory with

a commutative associative multiplication with unit. For X 1 2 (mod 4) Araki and Toad [6] constructs an

Iadmissible" _{multiplication}

hi(X; Z/1) 0 hd(Y; Z/L) -> hi+J(X A Y; Z/L).

h 1+2(X _{A M ) h J+2(Y A} +

h

(X A

+ i+J+(X A + ~1+j+4 (X A 2 i+j+2(X A

02

h

iJ+2(xA

Mz A Y A M) Y A Mt A ML) Y A ML A S2 ₎ Y A ML) - hi+J(X A Y;

where the first map is multiplication in h , the second map interchange of Mx and Y, the third map is induced by a certain map ML A S2 + Mt A M,, and the fourth map is the inverse of the double suspension.

Remark. We shall only need the case where L is a power of a prime t # p. Since we want to pass to t-adic coefficients t = lim Z/ta, _{we can aVoid the} unpleasant behavior of products in h*(-; Z/L) for L 1 2 (mod 4).

52.

M) Bi (X; Z/0) 0 hj(Y; Z/O)

-I-,

We extend the definition (4.4) of $ to

degrees < - 2 so that the following diagram commutes

R (-; Z/X) - -> (-; Z/L)

-I-2 _{-- 2}

k - A S--2; Z/X) .> 2 A -- 2; Z/ )

denotes suspension.

Lemma 4.7. If X 9 2 (mod 4) the diagrams

Z/1)

*

kF(Y; Z/ ) - kF

k i(X; Z/X) e k6(Y; Z/) > kC (X

with i < - 2, J _ - 2 are commutative,

Proof: If i

.J

= - 2 this follows immediately from corollary 4.3 and the definition of the

products V.. The general case follows from the Here a

k F(X; A Y; Z/1)

1$

54.

commutativity of the products V with suspension

([6] thm. 5.5),

Theorem 5.5 of [6] shows also that the reductions mod X

* *

Pj : kF( - ) + k

* *

PX : kc( - ) k

are multiplicative stable

Z/Z)

Z/L)

operations. From the known structure of kc(SO) and the proof of proposition 4.5 we get

(4.8) _{k C(S ; Z/x) =} (* 0 _Z/i[si]

-W2

₀

where S = e k- (S ) is the reduction mod I of the Bott class 8 e k2 (s) K(S )

The standard multiplicative stable operation

* *

k(- ) +~ K ( - )

-'i-i.

with Z/L-coefficients and we get a commutative diagram (we think here of K as a Z-graded theory)

kC( -> K

~1~ 1,

k (-; Z/L) > K (-; Z/2)

where the horizontal maps may be thought of as the localization [E I]

Lemma 4.9. Let k be prime to p and k ! 2 (mod 4). Then the composition

-. PX -2 z/)~ 2 -K 2(;z)

k-2( - k (-; Z/)+ k- (-; Z/) + K-2(-; Z/)

can be extended to a multiplicative stable operation

k(- ) K*(-; Z/)

Proof,. We can extend uniquely to degrees < 2 to get

56.

$ :+ (-; Z/A) (i < - 2)

which commutes with suspension and is multiplicative in degrees < - 2 (lemma 4*7).

Let

be the inverse image of X c k 2 (SO; Z/L) under the isomorphism (proposition 4.5)

~2 (S0; Z/t) -+ 2(S0 Z/)

Note that by proposition 4.5, lemma 4.7 and (4.8), we have

k(30 ; Z/X) = Z/1[ ]

In particular the coefficient ring of kF( ; Z/9) is associative, and a is a well defined element of k-2r(sO; Z/1) _{for dny r > 0.}

We now define to be the composition

F - ) - -> k1 (; 'Z/L) *.> kF-2(i.-; Z/L)

~-r

o

P.

-iiX-2r2

+ Ki-2r(.; Z/x) (-; Z/A)

where r > 0 is chosen, so that i 2r < - 2. In order to prove that this is independent of the choice of r we use the fact the associative law

(xy)z = x(yz)

for three elements i k (X; Z/L) holds, when at least one of the three elements is the reduction mod X of an element in kF(X) ([6] thm. 5.5). Using this and

lemma 4.7 together with the fact that multiplication

-**

in K (-; Z/A) is associative and commutative ([6] thm. 10.7), we get r-l W, r+lP (x)) . -r-l r r-$ ) W( p rC (x)) a ~r I 00 zS$S)$Sr P(x)) 1) 1 a

5'.

which proves the independence of r. Now let x e k F(X), y e k F(Y).

$ (x) = 0r O$t(xy) = -r-sO r+sPX) for suitable r $X(x)$t(y) and s. We get = 0rr-s s = r-s 0(0 rP(x)1{OsPz(y)

by lemma 4.7. In order to prove that , is multiplicative, we must prove that

Sr+s 5_{PR(xy) w{orPt(x)}

(s

Since

P~t XY)

Then

59.

we can use the above mentioned "quasi associative" law to reduce this to proving that

Sa pj(x) = pt(x)as"

It is obvious from pt'position 3.5, that 2 , S4 ) S2 induces the zero map on kF. This implies by thm. 7.4 of [6] that the multiplication in k ; Z/X) is

commutative. This proves that is multiplicative, and it is clearly stable.

Remark. In the case, where X L + 1 (mod 6) it is proven in [6] that the multiplication in

A#*

kF(-; Z/L) is commutative and associative, and the above proof becomes much simpler.

For a generalized cohomology theory h with product y, one can define a product

R hi(X; Z/X)

e

hJ(Y) +-p- hi+J(X A Y; Z/L)

60.

hi+2(X A Mx ) e h (Y) hi+j+2(X A M A Y)

+ _{i+J+2(X A Y A ML),}

where the first map is the product in h , and the second map is induced by interchange of Mt and Y. An

admissible multiplication V. in h (-; Z/L)

satisfies

1R = l

It follows that we for x e ki(X) have

(4.10) yx(x) = $-r, r @Lx0

where r > 0 and i - 2r < - 2.

Lemma 4.11. The following diagram is commutative

k( - ) kC( ) K (

{t)

..L *

Proof. For x e kF(X) we get corollary 4,3 and the definition of

$yROX 06 X) as a consequence of = P R((8R

*

= yR

6O

= P (C$x)) * S p $(x) By (4,10) we have $*(x) " 0 6X) =t OW *()

If X divides n and h is a generalized cohomology theory one can define reduction mod I

~W* ~ *

h (-; Z/n) * h (-; Z/L)

as the operation induced by the map T : M + Mn in the diagram

62.

I

i S > S ->M -> S2

-n

I

i S --- > S -- > Mn_ 2 Clearly

PnR

Pn

= Pi and %,L Pm,n = Pm,n if n divides m.

Lemma 4.,12. Let k and the prime number t # p with

n be powers (# 2) of L dividing n.

Then the diagram

OW*

K (-; Z/n)

k(-

)n,

K (-; Z/L)

63.

is commutative.

Proof_{4, It follows immediately from the definitions,}

that the diagram

k (-; Z/n)

(4.13) PnA pn, L (i < - 2)

kI(_{kF(-; Z/Z-)}

is commutative in degrees < - 2, and hence that the diagram of the lemma is commutative in degrees < - 2.

Since r kE2r (SO; Z/A) is the deduction mod Z

of r 2r k ), we have

an, n x

in k -theory. It follows from (4.13) that this also holds for k Ftheory. Now consider an element

x E ki(X) and choose r > 0, so that i - 2r < - 2.

We do the following computation, where the property

I"

k Z/n) C

0"

64.

P "R ' "R(Pn,X 0 I1)

of the multiplication yR is used:

= 'n "-R~ ('Rn

= x.

=n

For a finite CW-complex with basepoint X we define K-theory with coefficients in the ring of t-adic numbers Zt by

* *

K (X; Zt) = lim K (X; Z/tn).

Since K (X; Z/tn) is a finite abelian group, and

Pa, g$n(x) X)

X)

X)

65.

inverse limit is exact on the category of inverse systems of finite abelian groups indexed over the natrual numbers, this is a cohomology theory on the

category of finite CW-complexes.

* A

Since Zt is torsion free K ( - ) t is a

cohomology theory. The maps

K ( Z +

K(

-

)

0

Z/tn + K (-; Z/tn)

define a map of cohomology theories

K( - ) Z + K(-; Zt),

which is easily seen to be an isomorphism on the coefficients. Hence it is an isomorphism of cohomology theories. There

is a natural product in K (-; Zt) induced by the products in K ( - ) and Zt, such that the operation

*

(4.14) K -; Z t) + K (-; Z/tn)

is stable and multiplicative.

Observe, that since every element of K*(X, Z/L)

66.

to (assuming X prime to p)

K ( - ) @ Z[-] -+ K*(-; Z/L)

p

and K (-; Z/Z) is naturally isomorphic to the cohomology theory we get by introducing Z/X-coefficients in

K ( - ) Z[].

p

Lemma 4.15. There exists a multiplicative stable operation

kF( - K ( - ) Z[-],

F np

which in degree 0 is the composition

k0(- ) -i- kgC( - ) = K( - ) -- > K

Proof. Let t be a prime different By lemma 4.12 we have an operation

( 0 Z[13

_p

frrom p.

k0 *

making the diagrams

kF

,tn

(t) * ^

K _{(-; Z/tn)}

commutative. Since (4.14) is stable and multiplicative we can conclude from lemma 4.9, that (4.16) is stable

and multiplicative.

The last term of the exact sequence

0 + Z z

+ Z + t /Z(t) b 0

is a vector space over Q, so we get the bottom exact sequence of the diagram

k

}

(t)

68.

where the cohomology theory on the right is the direct sum of copies of H (-; Q) with shifts in degree. Lemma 4.11 implies that $(t) factors through

K ( - ) S Z in degree 0, and hence the composition

(4.17) kF( - ) +K - ) S Zt/Z(t)

is zero in degree 0. This factors through k - ) Q which as a consequence of proposition 3.5 is isomorphic

to H*(-; Q). Since the only stable operations in

H (-; Q) are rational multiples of the identity (4.17) is zero, and we can factor *(t) through an operation

* *

(4.18) kF( - ) -+ K ( - ) 9Z

which necessarily must be stable and multiplicative. In degree 0 (4.18) is the composition

iFO( - ) -> k0( - ) -=K( - ) + K( - ) 0 Z

When we apply the functor - 0 Q to (4.18), we get a stable operation

69.

* * *

H (-; Q) kF( - ) Q K ( - )

Q

which in degree 0 is independent of t. Since this operation is uniquely determined by its degree 0 part,

the composition

k*( - ) + K ( - )

Z

(t) - ) Q

is independent of t. This implies the existence of a unique operation

kF( - ) K( - ) Z[]

F p

making the diagrams

kF( - ) - > K ( - ) @ Z[ ]

F p

K ( - ) S Z(t)

with t # p commutative. This operation clearly has the required properties.

'10 .

Since kC - ) Z[ is the -1 - connected

*

1

covering theory of K ( - ) 0 Z[-1], the operation of

p

lemma 4,15 has a unique lifting to a multiplicative stable operation

(4.19) : k*( - ) + kg( - ) Z[1]

F C p

This proves the existence part of theorem 3.6 in the case, where F is the algebraic closure of a finite field. For a finite subfield F' of F we get the existence by composing with the multipLicative stable

* *

operation k - k F

F' F

Remark. It follows easily from proposition 4.5 that (4.19) induces an isomorphism of the mod X theories

k(; Z/X) +kC- Z/X)

for any L > 2 prime to p. The results of Araki and Toda [6] show that kC-; Z/X) always has an admissible associative multiplication. This can be

transported to an admissible associative multiplication in k*(-' Z/L), even though the criteria given in [6]

11 0.

are too weak to conclude the existence of an admissible associative multiplication in kF(-; Z/L).

We proceed to prove the uniqueness in theorem 3.6. We let F be a finite field or the algebraic closure of a finite field and 7 the algebraic closure of F. Assume that

(4920) : kF( - ) +kC( - ) eZ[1]

is a multiplicative stable operation extending the given $ in degree 0.

Let X = tn # 2 be a power of a prime t # p. If F has q elements we choose r > 1 so that

q = 1 (mod X).

From (3.3), proposition 3.5 and the universal coefficient theorem we see that the inclusion F + 7 induces an

isomorphism

72.

We let 0 0 k 2r(S0 ; Z/) be the inverse image of

the element r kp2r (S; Z/1), In the case F =

we set r w 1 and (r) = . Then the operation

k(; Z/X) +k(- Z/0)

induced by (4,20) maps O into _p,(cr) . kC(-; Z/0)

and is stable and multiplicative with respect to

suitable admissible products. The composition

k (-; Z/L) + k*(-; Z/k) + K*(-; Z/1)

is determined in degrees < - 2 by the given

*

in

degree 0. In degrees > 1 it is determined by the

condition that it commutes multiplication by O r) and

a . It follows that the composition

k ( - ) + k *( - ) 0 Z[1] + K*( -

)

Z[] + K*(-; Z/L)

FCp p

is uniquely determined. By the method used in the proof

of lemma. 4,15, we conclude that the composition

is uniquely determined. Since k is -1 - connected we have a unique lifting to k*( - ) 0 Z[1.]. This

t pp

*j' L1*

5. Extension to Infinite CW-complexes, For a prime p we let

Q(C,

i)[1]

p

denote the localization of Q(C, i) in the sense of Sullivan [17]. In particular we have a homotopy equiValence

Q(C,

1)[i1 = BU[ ]

p p X p0

This defines a -1 - connected f-spectrum Q(C)[ ] and a map of spectra

9(C) -+ g(C)[ ]

There is a natural multiplication on Q(C)[1], such that we get a nultiplicative stable operation

k*( - ) -+ kC[!](

of cohomology theories defined on all CW-complexes. For a finite CW-complex X we have an isomorphism

7

5.

k c(X)

e

Z[ ]

=

o[

](X.

This does not hold for infinite CW-complexes as can be seen from the example X = K(Z/p, 1).

The next theorem shows that theorem 3,6 can be extended to the category of all CW-complexes.

Theorem 5.1. Ket F be a finite field or the

algebraic closure of a finite field, and let p = char F. For every i > 0 the natural transformation of theorem 3.6

S: - ) - ) Z[]

defined on the category of finite CW-complexes, is represented by a unique homotopy class

Q(F, i) + Q(C, 1)[]

p

7 U Q(F11 Q(,i)

A

Q(F,

J)

- V C, 1)[5]

A

Q(C, J)[-] p p

1/

1V

Q(F, i+j) *>Q(C, i+j)[ _p71

are homotopy commutative.

Immediate consequences are:

Corollary 5.2. Theorem 3.6 generalized to the

category of all CW-complexes.

Corollary

5.3

The map

B F x _Fp Z l > BU[1] x Z[1]

p

is an infinite loop map.

In the case of a finite field F, we shall use the following lemma.

Lemma 5,4. Let X be a CW-complex with finite skeletons Xm and j an integer, such that

H (X; 0 for i j

77.

Then we have an isomorphism

k*[ r(X) klim[ ](XI

Op I- Cp Ii (m

m

Proof. In the AtIyah-Hirzebruch spectral eequence for kC[ ](X) the term

Es.%t ₂ = s _{s(X, kC(pt) 0} k _Z[p]))

is a finite abelian group for s + t = J - 1, Hence

EsIt = ES,t

r+1. r (s+t = J-1)

for r > r(st), where r(s, t) is an integer depending on s and t. By a well known argument (see Adams

[1] or 2 of Hodgkin [11]) the inverse system

.. k k

]1.1((Xm) [ p Xm'

70.

lim kC [1 (X ) = 0

4- C p m

m

-and we can apply Milnor's lemma [13] to get the isomorphism. Proof of 5.1 for F finite. By (3.3) and Serre

theory the map

Q(F, i) - K(Z, i)

induces an isomorphism on cohomology with rational coefficients. Hence

H (Q(F, 21); Q)

is a polynomial algebra on a 21-dimensional generator, and

H *(Q(F, 21-1);

Q)

lb an exterior algebra on a generator of dimension 21-1. Lemma 5.4 gives us the isomorphisms

rlim

kC[

1

QF

i)

A

Q(F,

J))

.lim k

p

[ (Q(F, _{1)m A Q(F, J)n)} Theorem 5.1 follows immediately.

Now le field

F

t F be the algebraic and consider F as th

closure of the prime e union of the finite fields _Fq, where q runs through the powers

Then Q(F, i) can be considered as the direct limit of _Q(Fq 1).

Lemma 5.5. For i > 0 and J > 0 we have isomorphisms

k$[ ](Q(F, i)) i) A Q(F, j)) = -lim k i[ .]CQ(F .4 C p ci qi 4- C p q q

Proof. Assume i is odd and let represent a generator of S - Q(F, - Z. The composition 79. [ 1 (i( i)) i)m) of p. i)) i) A Q(Fq,

j))

i) kI [' ](Q(F, ki+ [ ](Q(F, f i(,Q(Fp, i))

80.

i)

Sr +en _{Q(F, g) +o Q(Fq}

represents a generator of i i(Q(Fq, i)) = Z, and it induces an isomorphism on H (-; Q). We form the cofiber

S . Q(F i) Y

which has trivial rational cohomology. Since

k l(S i) = 0, we get an epimorphism of inverse systems

- [4ki-l _{(Q(F ))}

By the vanishing of lim2 we have an epimorphism

over the present index set,

lim 1 kqiJ[211 ) + li1m 1k i-l[ (F i).

I- C p q 4- C pq$

q q

The E2term of the Atiyah-Hirzebruch spectral ~* 1

sequence for kc[](Yq) consists of finite abelian groups. This implies, that the skeleton filtration defines a

compact topology on k 31A.](Y ). It follows that C p q Urm ' iqlK(y) = 0 4- C p

q

q and hence lim kil[I](Q(F 1)) = 0. q

Milnor's lemma gives us the first isomorphism of the lemma for i odd.

For i even we use the cofibration sequence

3 i+l -+ Q(F q 1) + Zq _,

where the first map is the loop of

F +e -)1 Q(Ft e+c)s

From the Atiyah-Hirzebruch spectral sequence is seen that

82.

k -1 [I +1S ) = 0. C p

Moreover Z. has trivial rational cohomology. The q

argument for i odd can now be carried through.

The last statement of the lemma can be proved in

the same manner using one of the spaces

Si A SJ, Si A ISJ +1 2 1+1 A 1P ~+1 Anj+ls

depending on the congruence classes of i and j mod 2. Proof of 5.1 for F algebraically closed. By

the first isomorphism of lemma 5.5, the homotopy classes

Q(Fq, 1) + Q(C, 1)[1] p

combine to a unique homotopy class

Q(F, i) +

Q(C,

i)[i]

p

The commutativity with multiplication follows from the second isomorphism of lemma 5.5. Finally the uniqueness follows from the uniqueness for finite fields.

O.)e

We remind the reader, that all the maps constructed here depend on- the choice of the embedding (1.3).

We can now prove the following:

Theorem 5.6. Let F be a finite field or the algebraic closure of a finite field. Then the Quillen map

BF * BU

can be delooped infinitely many times.

Proof. The spaces BF and BC = BU are the bottom spaces of 0-connected P-spectra B and C From the fibrations

K(Z, 1-1) -+ B + Q(F, )

K(Z[I], i-1) +o- BC(C 1

p C p

is seen that the maps

Q(F,

i) + Q(C, i)[1 ]

p