Algebraic K-theory of rings from a topological viewpoint Dominique Arlettaz Abstract. Because of its strong interaction with almost every part of * *pure math- ematics, algebraic K-theory has had a spectacular development since it* *s origin in the late fifties. The objective of this paper is to provide the basic * *definitions of the algebraic K-theory of rings and an overview of the main classical theo* *rems. Since the algebraic K-groups of a ring R are the homotopy groups of a topolo* *gical space associated with the general linear group over R , it is obvious that m* *any general re- sults follow from arguments from homotopy theory. This paper is essent* *ially devoted to some of them: it explains in particular how methods from stable hom* *otopy theory, group cohomology and Postnikov theory can be used in algebraic K-theor* *y. Table of contents 0.Introduction p. 1 1.The origins of algebraic K-theory p. 2 2.The functors K1 and K2 p. 4 3.Quillen's higher K-groups p.10 4.The product structure in algebraic K-theory and the K-theorypspectrum* *.16 5.The algebraic K-theory of finite fields p.21 6.The Hurewicz homomorphism in algebraic K-theory p.25 7.The Postnikov invariants in algebraic K-theory p.31 8.The algebraic K-theory of number fields and rings of integersp.39 9.The algebraic K-theory of the ring of integers Z p.43 10.Further developments p.53 References p.54 0. Introduction Algebraic K-theory is a relatively new mathematical domain which grew up at the* * end of the fifties on some work by A. Grothendieck on the algebraization of category theory (see Section 1* *). The category of finitely generated projective modules over a ring R was actually in the center of the pr* *eoccupations of the first K-theorists because of its relationships with linear groups which play a crucia* *l role in almost all subjects in mathematics. Later, J. Milnor and D. Quillen introduced a very general notio* *n of algebraic K-groups Ki(R) of any ring R which exhibits some properties of the linear groups over R * *(see Sections 2 and 3). Thus, in some sense, algebraic K-theory is a generalization of linear algebra o* *ver rings! 1 The abelian groups Ki(R) are homotopy groups of a space which is canonically as* *sociated with the general linear group GL(R) , i.e., with the group of invertible matrices, over the ring* * R . Therefore, several methods from homotopy theory produce interesting results in algebraic K-theory of rings* *. The objective of the present paper is to describe some of them. This will give us the opportunity to introdu* *ce the definition of the groups Ki(R) for all integers i 0 (in Sections 1, 2 and 3), to explore their structur* *e (in Section 4) and to present classical results (in Sections 5 and 8). Moreover, the second part of the paper* * (Sections 6, 7, 8 and 9) is devoted to more recent results. Of course, this is far from being a complete list of topological arguments used* * in algebraic K-theory and many other methods also provide very nice results. On the other hand, if, inste* *ad of looking at the category of finitely generated projective modules over a ring, we apply the same ideas t* *o other categories, we also get interesting and important developments of algebraic K-theory in various directi* *ons in mathematics. If the reader wants to get a better and wider understanding of algebraic K-theory, he * *should consult the standard books on this subject (see for example [19], [27], [29], [58], [65], [78], [83]* *, [90]; a historical note can be found in [22]). Since our goal is to show how algebraic topology (in particular homotopy theory* *) can be used in algebraic K-theory, we assume that the reader is familiar with the basic notions and resu* *lts in algebraic topology, homotopy theory and homological algebra (classical references include [2], [37]* *, [47], [50], [51], [79], [89], [94], [100]). Throughout the paper, all maps between topological spaces are supposed to be co* *ntinuous pointed maps and all rings are rings with units. 1. The origins of algebraic K-theory The very beginning of algebraic K-theory is certainly due to some general consi* *derations made by A. Grothendieck. He was motivated by his work in algebraic geometry and introduced* * the first K-theoretical notion in terms of category theory. His idea was to associate to a category C a* *n abelian group K(C) defined as the free abelian group generated by the isomorphism classes of objects of C * *modulo certain relations. Since Grothendieck's mother tongue was German, he chose the letter K for denoti* *ng this group of classes ( K = Klassen) of objects of C . This group K(C) was the first algebraic K-grou* *p: it is now called the Grothendieck group of the category C. Let us explain in more details the definition of a Grothendieck group by lookin* *g at two classical examples. First, let R be a ring and P(R) the category of finitely generated projective R* *-modules (see [27], Sections VII.1 and IX.1, [65], Chapter 1, [78], Section 1.1, [83], Chapter 2, or [90], p* *.1). Definition 1.1. For any ring R , the Grothendieck group K0(R) is the quotient o* *f the free abelian group on isomorphism classes [P] of finitely generated projective modules P 2 P(R) by* * the subgroup generated by the elements of the form [P Q] - [P] - [Q] for all P , Q in P(R) . Thus, every element of K0(R) can be written as a difference [P] - [Q] of two ge* *nerators and one can easily check (see [65], Lemma 1.1, [83], p.9, Proposition 2, or [90], p.1) that* * two generators [P] and [Q] are equal in K0(R) if and only if there is a free R-module Rs such that P Rs~=Q R* *s (in that case, the R-modules P and Q are called stably isomorphic). 2 A homomorphism of rings ' : R ! R0 induces a homomorphism of abelian groups '* : K0(R) -! K0(R0) which is defined as follows. If P is a finitely generated projective R-module, * *there is an R-module Q and a positive integer n such that P Q ~=Rn and consequently, (R0R P)(R0R Q) ~=R0R Rn* * ~=(R0)n : in other words R0RP is a finitely generated projective R0-module. Therefore, let u* *s define for all [P] 2 K0(R) '*([P]) = [R0R P] 2 K0(R0) ; K0(-) turns out to be a covariant functor from the* * category of rings to the category of abelian groups. The main problems in the study of the group K0(R) are the following two questio* *ns which express the difference between the classical linear algebra over a field and the algebraic * *K-theory which concerns any ring R : is every finitely generated projective module over R a free R-module? * * and is the number of elements in a basis of a free R-module an invariant of the module? If both ques* *tions would have a positive answer, then the group K0(R) would be infinite cyclic, generated by the class [* *R] of the free R-module of rank 1. This is of course the case if R is a field, but also for other classes * *of rings. Theorem 1.2. If R is a field or a principal domain or a local ring, then K0(R) * *~=Z , generated by the class of the free R-module of rank 1. Proof. See [65], Lemma 1.2, or [78], Sections 1.1 and 1.3. * * _|_| The main interest of algebraic K-theory in dimension 0 is the investigation of * *projective modules which are not free. For instance, let us look at Dedekind domains, in particular at r* *ings of algebraic integers in number fields. Theorem 1.3. If R is a Dedekind domain, then K0(R) ~=ZC(R) , where C(R) denotes* * the class group of R . Proof. See [65], Chapter 1, or [78], Section 1.4. * * _|_| The second classical example of a Grothendieck group is given by the topologica* *l K-theory which was introduced by A. Grothendieck in 1957 (see [33]) and developed by M.F. Atiyah a* *nd F. Hirzebruch (see [20] and [19]). If X is any compact Hausdorff topological space and F = R or C , let* * us denote by V(X) the category of F-vector bundles over X . Definition 1.4. For any compact Hausdorff topological space X , the Grothendiec* *k group K0F(X) is the quotient of the free abelian group on isomorphism classes [E] of F-vector bundl* *es E 2 V(X) by the subgroup generated by the elements [E G] - [E] - [G] for all E , G in V(X) , w* *here is written for the Whitney sum of vector bundles. The abelian group K0F(X) is called the topologic* *al K-theory of X (see [19] or Part II of [51] for more details). Again, every element of K0F(X) is of the form [E] - [G] where E and G are two v* *ector bundles and [E] = [G] if and only if there is a trivial F-vector bundle L such that E L ~=* *G L ( E and G are then called stably equivalent). If f : X ! Y is a continuous map between two compac* *t Hausdorff topological spaces, and if E -p! Y is an F-vector bundle over Y , then f*(E) = {(x; e) 2 X* * x E | f(x) = p(e)} together with f*(p) : f*(E) ! X given by f*(p)(x; e) = x defines an F-vector bu* *ndle over X. Thus, the map f induces a homomorphism of abelian groups f* : K0F(Y ) -! K0F(X) 3 and it turns out that K0F(-) is a contravariant functor from the category of co* *mpact Hausdorff topological spaces to the category of abelian groups. Remark 1.5. It is not difficult to show that the group K0F(X) splits as K0F(X) ~=Z eK0F(X) ; where Ke0F(X) , the reduced topological K-theory of X , is the kernel of the ho* *momorphism K0F(X) ! Z which associates to each vector bundle its rank. For example, the calculation of Ke0F(X) in the case where X is an n-dimensional* * sphere Sn is provided by the celebrated Theorem 1.6. (Bott periodicity theorem) n (a) eK0C(Sn) ~= Z; if n is even, 0 if n is odd. 8 >>>Z; if n 0 mod 8, >>>Z=2;if n 1 mod 8, >>>Z=2;if n 2 mod 8, <0; if n 3 mod 8, (b) eK0R(Sn) ~=> >>>Z; if n 4 mod 8, >>>0; if n 5 mod 8, >>:0; if n 6 mod 8, 0; if n 7 mod 8. Proof. See [35] or [51], p.109 and Chapter 10; an alternative proof may be foun* *d in [46]. _|_| In fact, there is a very strong relationship between topological and algebraic * *K-theory: Theorem 1.7. (Swan) Let X be any compact Hausdorff topological space, F = R orC* * , and R(X) the ring of continuous functions X ! F . There is an isomorphism of abelian groups K0F(X) ~=K0(R(X)) : Proof. See [93] or [78], Theorem 1.6.3. * * _|_| Remark 1.8. It is also possible to describe K0(R) in terms of idempotent matric* *es over R (see for instance [78], Section 1.2). This approach is the first sign of the central role played * *by linear groups in algebraic K- theory: it will become especially important for higher K-theoretical functors i* *n the next sections. 2. The functors K1 and K2 One of the main objects of interest in linear algebra over a ring R is the gene* *ral linear group GLn(R) consisting of the multiplicative group of n x n invertible matrices with coeffi* *cients in R . In order to look at all invertible matrices of any size in the same group, observe that GLn(R) m* *ay be viewed as a subgroup of GLn+1(R) via the upper left inclusion A 7! A0 01 and consider the direct l* *imit 1[ GL(R) = lim-!GLn(R) = GLn(R) n n=1 4 which is called the infinite general linear group. Algebraic K-theory is essent* *ially the study of that group for any ring R . To begin with, let us investigate the commutator subgroup of G* *L(R) . Definition 2.1. Let R be any ring, n a positive integer, i and j two integers w* *ith 1 i; j n , i 6= j , and an element of R ; let us define the matrix ei;jto be the n x n matrix with* * 1's on the diagonal, in the (i; j)-slot and 0's elsewhere: such a matrix is called an elementary mat* *rix in GLn(R) . Let En(R) denote the subgroup of GLn(R) generated by these matrices and let 1[ E(R) = lim-!En(R) = En(R) ; n n=1 where the direct limit is taken via the above upper left inclusions. The group * *E(R) is called the group of elementary matrices over R . An easy calculation produces the next two lemmas (see [78], Lemma 2.1.2 and Cor* *ollary 2.1.3). Lemma 2.2. The elementary matrices over any ring R satisfy the following relati* *ons: (a) eijeij= e+ij, 8 ><1* * ; if j 6= k, i 6= l, (b) The commutator [eij; ekl] = eijekl(eij)-1(ekl)-1 satisfies [eij; ekl] = > e* *il; if j = k, i 6= l, : e- * *kj ; if j 6= k, i = l, for all and in R . (Notice that there is no simple formula for [eij; ekl]* * if j = k and i = l .) Lemma 2.3. (a) Any triangular matrix with 1's on the diagonal and coefficients in R belong* *s to the group E(R) . (b) For any matrix A 2 GLn(R) , the matrix A0 A0-1 is an element of E2n(R) . The main property of the group of elementary matrices E(R) is the following. Lemma 2.4. (Whitehead lemma) For any ring R , the commutator subgroups [GL(R); * *GL(R)] and [E(R); E(R)] are given by [GL(R); GL(R)] = E(R) and [E(R); E(R)] = E(R) : Proof. Because of Lemma 2.2 (b) every generator eijof E(R) can be written as th* *e commutator [eij; e1jl] . Thus, one gets [E(R); E(R)] = E(R) and the inclusion E(R) [GL(R); GL(R)] . In * *order to prove that [GL(R); GL(R)] E(R) , observe that for all matrices A and B in GL(R) , one has ABA-1B-1 0 AB 0 A-1 0 B-1 0 0 1 = 0 B-1A-1 0 A 0 B and therefore this matrix belongs to E2n(R) according to Lemma 2.3 (b). * * _|_| Remark 2.5. Remember that a group G is called perfect if G = [G; G] or, in othe* *r words, if the abelian- ization Gab of G is trivial, or, in homological terminology, if H1(G; Z) = 0 (s* *ee [37], Section II.3, or [50], Section VI.4). Lemma 2.4 asserts that E(R) is a perfect group for any ring R . The discovery of that relationship between GL(R) and E(R) was the first step to* *wards the understanding of the linear groups over a ring R : in 1962, it gave rise to the following def* *inition (see [29], Chapter 1, [65], Chapter 3, or [78], Definition 2.1.5). 5 Definition 2.6. For any ring R , let K1(R) = GL(R)=E(R) = GL(R)ab: A ring homomorphism f : R ! R0 induces obviously a homomorphism of abelian grou* *ps f* : K1(R) ! K1(R0) and K1(-) is a covariant functor from the category of rings to the categ* *ory of abelian groups. Remember that the abelianization of any group G is isomorphic to its first homo* *logy group with integral coefficients H1(G; Z) (see [37], Section II.3 or [50], Section VI.4). Corollary 2.7. K1(R) ~=H1(GL(R); Z) . If R is commutative, the determinant of square matrices is defined and we may c* *onsider the group SLn(R) of nxn invertible matrices with coefficients in R and determinant +1 , and the * *infinite special linear group 1[ SL(R) = lim-!SLn(R) = SLn(R) : n n=1 This provides the extension of groups 1 -! SL(R) -! GL(R) det-!Rx -! 1 ; where Rx = GL1(R) is the group of invertible elements in R . Observe that E(R) * *is clearly a subgroup of SL(R) since any elementary matrix ei;jhas determinant +1 . By looking at the* * above extension and taking the quotient by E(R) , one gets the short exact sequence of abelian grou* *ps 1 -! SL(R)=E(R) -! K1(R) -! Rx -! 1 which splits since the composition of the inclusion of Rx = GL1(R) = GL1(R)=E1(* *R) into the group GL(R)=E(R) = K1(R) with the surjection K1(R) ! Rx is the identity. Therefore, w* *e can introduce the functor SK1(-) and obtain the next theorem. Definition 2.8. For any commutative ring R , let SK1(R) = SL(R)=E(R) : Theorem 2.9. For any commutative ring R , K1(R) ~=Rx SK1(R) . Consequently, it is sufficient to calculate SK1(R) in order to understand K1(R)* * . Let us first mention the following vanishing results. Theorem 2.10. If R is a field or a commutative local ring or a commutative eucl* *idean ring or the ring of integers in a number field, then SK1(R) = 0 and the determinant induces an isom* *orphism K1(R) ~=Rx : Proof. See [78], Sections 2.2 and 2.3. * * _|_| Example 2.11. K1(Z) ~=Z=2 = {1; -1} , K1(Z[i]) ~=Z=4 = {1; i; -1; -i} , and K1* *(F[t]) ~=Fx for any field F . However, SK1(R) does not vanish for all commutative rings R . Here is a result* * on K1 for Dedekind domains. 6 Definition 2.12. Let R be a commutative ring, and a and b two elements of R suc* *h that Ra+Rb = R . If c and d are elements of R such that ad - bc = 1 , then the class of acbd * *in SK1(R) does not hbi depend on the choice of c and d (see [78], Theorem 2.3.6): it is denoted by a * * and called a Mennicke symbol. If R is a Dedekind domain, it is possible to prove that K1(R) is generated by t* *he image of GL2(R) in K1(R) = GL(R)=E(R) (see [78], Theorem 2.3.5). This implies the following result. Theorem 2.13. If R is a Dedekind domain, then the group SK1(R) consists of Menn* *icke symbols. Remark 2.14. In that case, SK1(R) is in general non-trivial. However, if R is a* * Dedekind domain such that R=m is a finite field for each non-trivial maximal ideal m of R , then S* *K1(R) is a torsion abelian group (see [78], Corollary 2.3.7). The next step in the study of the linear groups over a ring R was made by J. Mi* *lnor and M. Kervaire in the late sixties when they investigated the universal central extension of the * *group E(R) (see [65] and [54]). Definition 2.15. Let G be a group and A an abelian group. A central extension o* *f G by A is a group H together with a surjective homomorphism ' : H!! G such that the kernel of ' i* *s isomorphic to A and contained in the center of H . A morphism from the central extension ' : H!! G * *to the central extension '0: H0!! G is a group homomorphism : H ! H0 such that ' = '0 . A central ext* *ension ' : H!! G of G is universal if for any central extension '0: H0!! G there is a unique mor* *phism of central extensions : H ! H0. Remark 2.16. This universal property implies clearly that any two universal cen* *tral extensions of a group G must be isomorphic. The main result on the existence of universal central extensions is provided by* * the following theorem. Theorem 2.17. (a) A group G has a universal central extension if and only if G is a perfect g* *roup. (b) In that case, a central extension ' : H!! G of G is universal if and only * *if H is perfect and all central extensions of H are trivial (i.e., of the form of the projection A * *x H!! H for some abelian group A ). Proof. See [65], Theorems 5.3 and 5.7. * * _|_| Remark 2.18. In fact, if G is a perfect group presented by R! F!! G (with F * *a free group), the proof of Theorem 2.17 (a) explicitly constructs a universal central extension '* * : [F; F]=[R; F]!! G and it turns out that the kernel of ' is (R \ [F; F])=[R; F] (see [65], Corollary 5* *.8). On the other hand, this corresponds to the Hopf formula for computing the second integral homology grou* *p of a group G (see [37], Theorem II.5.3 or [50], Section VI.9) and we immediately obtain the following c* *onsequence: if ' : H!! G is the universal central extension of the perfect group G , then H2(G; Z) ~=ker' : Since the group of elementary matrices E(R) is perfect for any ring R according* * to Lemma 2.4, it has a universal extension which can be described as follows. 7 Definition 2.19. Let R be any ring and n an integer 3 . The Steinberg group St* *n(R) is the free group generated by the elements xijfor 1 i; j n , i 6= j , 2 R , divided by the re* *lations (a) xijxij= x+ij, ae1 , if j 6= k and i 6= l, (b) [xij; xkl] = xil , if j = k and i 6= l. Remark 2.20. The relation (a) implies that (xij)-1 = x-ij. It then follows from* * (b) that for k 6= j , k 6= i , one has x-kixijxkix-ij= x-kj and consequently xijxkix-ijx-ki= xkix-kjx* *-ki= x-kj since [xki; x-kj] = 1 by (b). In other words, [xij; xkl] = x-kj if j 6= k and i = l . Proposition 2.21. For any ring R and any integer n 3 , the Steinberg group Stn* *(R) is a perfect group. Proof. It is obvious that [Stn(R); Stn(R)] = Stn(R) since every generator xij i* *s a commutator by the equality xij= [xis; x1sj] . * * _|_| There is clearly a group homomorphism Stn(R) ! Stn+1(R) sending each generator * *xijof Stn(R) to the corresponding generator xijof Stn+1(R) and it is therefore possible to define t* *he infinite Steinberg group St(R) = lim-!nStn(R) . For any ring R and any integer n 3 , there is a surjective homomorphism ' : St* *n(R)!! En(R) defined on the generators by '(xij) = eijwhich induces a surjective homomorphism ' : St(R)!! E(R) : The infinite Steinberg group St(R) plays an important role for the group E(R) b* *ecause of the following result. Theorem 2.22. The kernel of ' is the center of the group St(R) and ' : St(R)!! * *E(R) is the universal central extension of E(R) . Proof. See [65], Theorem 5.10, or [78], Theorems 4.2.4 and 4.2.7. * * _|_| This gives rise to the definition of the next K-theoretical functor. Definition 2.23. For any ring R , let K2(R) be the kernel of ' : St(R)!! E(R) .* * Notice that K2(R) is an abelian group which is exactly the center of St(R) . A ring homomorphism f : R ! R0 induces group homomorphisms St(R) ! St(R0) and E* *(R) ! E(R0) and consequently a homomorphism of abelian groups f* : K2(R) ! K2(R0) . Thus, K* *2(-) is a covariant functor from the category of rings to the category of abelian groups. Remark 2.24. In fact, the relations occuring in the definition of the Steinberg* * group St(R) correspond to the obvious relations of the group of elementary matrices E(R) . However, E(* *R) has in general more relations and the group K2(R) measures the non-obvious relations of E(R) . From* * that viewpoint, the knowledge of K2(R) is essential for the understanding of the structure of the g* *roup E(R) . Remark 2.18 immediately implies the following homological interpretation of the* * functor K2(-) . 8 Corollary 2.25. For any ring R , K2(R) ~=H2(E(R); Z) . Remark 2.26. The definitions of K1(R) and K2(R) show the existence of the exact* * sequence 0 -! K2(R) -! St(R) -! GL(R) -! K1(R) -! 0 ; where the middle arrow is the composition of the homomorphism ' : St(R)!! E(R)* * with the inclusion E(R) ,! GL(R) . For the remainder of this section, let us assume that R is a commutative ring. * *Consider the above ho- momorphism ' : St(R)!! E(R) . If x and y belong to E(R) , one can choose elem* *ents X and Y in St(R) such that '(X) = x and '(Y ) = y . Of course, X and Y are not uniqu* *e; however, the commutator [X; Y ] is uniquely determined by x and y , because for any a and b * *in ker' one has [Xa; Y b] = XaY ba-1X-1b-1Y -1= XY X-1Y -1= [X; Y ] since a and b are central. * *Consequently, we can look at the commutator0[X;1Y ] 2 St(R)0and observe1that '([X; Y ]) = 1 if x* * and y commute in E(R) . u 0 0 v 0 0 Thus, we choose x = @ 0 u-1 0A and y = @ 0 1 0 A in E3(R) (see Lemma 2.3 (b)* *), where u 0 0 1 0 0 v-1 and v are invertible elements in R . Since R is commutative, the elements x and* * y commute and we get an element [X; Y ] of K2(R) . Definition 2.27. Let R be a commutative ring. For all u and v in Rx , the Stein* *berg symbol of u and v is the element {u; v} = [X; Y ] 2 K2(R) , where X and Y are chosen as above. Lemma 2.28. For all u , v and w in Rx , the Steinberg symbols have the followin* *g properties in K2(R) : (a) {u; v} = {v; u}-1, (b) {uv; w} = {u; w}{v; w} , (c) {u; vw} = {u; v}{u; w} , (d) {u; -u} = 1 , (e) {u; 1 - u} = 1 , when (1 - u) 2 Rx . Proof. See [65], Chapter 9, or [78], Lemma 4.2.14 and Theorem 4.2.17. * * _|_| Example 2.29. Let R be the ring of integers Z . Then Zx has only two elements: * *1 and -1 . By Lemma 2.28, it is clear that {1; -1} = {-1; 1} = {1; 1} = 1 and that {-1; -1} is at m* *ost of order 2 . In fact, the group K2(Z) is generated by the Steinberg symbol {-1; -1} and K2(Z) ~=Z=2 (see * *[65], Chapter 10). Many results have been obtained on K2(R) in the case where R is a field. The fi* *rst one asserts that K2 of a field is generated by Steinberg symbols and that the above properties (a) * *and (d) in Lemma 2.28 follow from the other three. Theorem 2.30. (Matsumoto) If F is a field, then K2(F) is the free abelian grou* *p generated by the Steinberg symbols {u; v} , where u and v run over Fx , divided by the relations* * {uv; w} = {u; w}{v; w} , {u; vw} = {u; v}{u; w} , {u; 1 - u} = 1 . Proof. See [65], Theorem 11.1, or [78], Theorems 4.3.3 and 4.3.15. * * _|_| If F is a finite field, the situation is much simpler because of the following * *vanishing result. 9 Theorem 2.31. If F is a finite field, then all Steinberg symbols in K2(F) are t* *rivial. Proof. See [65], Corollary 9.9, or [78], Corollary 4.2.18. * * _|_| We then may deduce a direct consequence of Theorems 2.30 and 2.31. Corollary 2.32. If F is a finite field, then K2(F) = 0 . Theorem 2.30 suggests a possible algebraic generalization of the functor K2(-) * *to higher dimensions: the Milnor K-theory which is defined as follows (see [64]). Definition 2.33. For any field F , the tensor algebra over Fx is 1M T*(Fx ) = Tn(Fx ); n=1 where Fx is considered as an abelian group and Tn(Fx ) = Fx_Z_Fx_Z_._.Z.Fx-z________": n copies In this algebra, one can consider the ideal I generated by all elements of the * *form u (1 - u) when u belongs to Fx . Then the Milnor K-theory of F is defined by KM*(F) = T*(Fx )=I : For each positive integer n , the elements of KMn(F) are the symbols {u1; u2; :* * :;:un} , with the ui's in Fx satisfying the following rules (additively written): (a) {u; -u} = 0 , (b) {u; 1 - u} = 0 , (c) {u; v} = -{v; u} , and KM*(F) has an obvious multiplicative structure given by the juxtaposition o* *f symbols. A lot of work has been done in the study and calculation of Milnor K-theory. Ho* *wever, we shall not discuss it here since the purpose of this paper is to study another generalization of K* *1 and K2, based on topological considerations: Quillen's higher K-theory. 3. Quillen's higher K-groups The main ingredient of the notions introduced in Section 2 is the investigation* * of the linear groups GL(R) , E(R) and St(R) . In order to generalize them and to define higher K-theoretical* * functors, the idea presented by D. Quillen in 1970 (see [70]) consists in trying to construct a topological * *space corresponding in a suitable way to the group GL(R) and studying its homotopical properties. Let us first di* *scuss a very general question on the relationships between topology and group theory. It is well known that one can associate with any topological space X its fundam* *ental group ss1(X) and with any discrete group G its classifying space BG which is an Eilenberg-MacLane spa* *ce K(G; 1) . This means that the homotopy groups of BG are all trivial except for ss1(BG) ~=G and impli* *es that the homology of the group G and the singular homology of the space BG are isomorphic: H*(G; A) * *~=H*(BG; A) for all coefficients A (see [37], Proposition II.4.1). Thus, the correspondence G ! BG * *and X ! ss1(X) fulfills 10 ss1(BG) ~=G , but it is in general not true that the classifying space of the f* *undamental group of a space X is homotopy equivalent to X : in other words, a topological space X is in gen* *eral not a K(G; 1) for some group G . Therefore, we are forced to introduce a new way to go from group* * theory to topology. This was essentially done by D. Quillen when he introduced the plus construction (se* *e [70], [59], Section 1.1, [3], Section 3.2, [29], Chapter 5, [48], [78], Section 5.2, or [90], Chapter 2). Theorem 3.1. Let X be a connected CW-complex, P a perfect normal subgroup of it* *s fundamental group ss1(X) . There exists a CW-complex X+P, obtained from X by attaching 2-cells an* *d 3-cells, such that the inclusion i : X ,! X+Psatisfies the following properties: (a) the induced homomorphism i* : ss1(X) ! ss1(X+P) is exactly the quotient map* * ss1(X)!! ss1(X)=P , ~= + * * + (b) i induces an isomorphism i* : H*(X; A) -! H*(XP ; A) for any local coeffici* *ent system A on XP , (c) the space X+Pis universal in the following sense: if Y is any CW-complex a* *nd f : X ! Y any map such that the induced homomorphism f* : ss1(X) ! ss1(Y ) fulfills f*(P) = 0* * , then there is a unique map f+ : X+P! Y such that f+ i = f . In particular, X+Pis unique up to hom* *otopy equivalence. Proof. The idea of the proof of this theorem is the following. We first attach * *2-cells to the CW-complex X in order to kill the subgroup P of ss1(X) ; then, we build X+Pby attaching 3-* *cells to the space we just obtained because X+Pmust have the same homology as the original space X . Obser* *ve that this creates a lot of new elements in the homotopy groups of X+Pin dimensions 2 . For details* *, see [59], Section 1.1, [29], Chapter 5, or [78], Section 5.2. * * _|_| The universal property of the plus construction implies the following assertion. Corollary 3.2. Let X and X0 be two connected CW-complexes, P and P0 two perfect* * normal subgroups of ss1(X) and ss1(X0) respectively and f : X ! X0 a map such that f*(P) P0. Th* *en there is a map f+ : X+P! X0+P0(unique up to homotopy) making the following diagram commutative: X ---f---! X0 ?? ? y i ?yi0 + X+P ---f---! X0+P0; and it is easy to check that the plus construction is functorial. Let us also mention the following important property. Lemma 3.3. If Xb is the covering space of X associated with the perfect normal * *subgroup P of ss1(X) , then there is a homotopy equivalence (Xb)+P' gX+P between (Xb)+Pand the universal cover gX+Pof X+P. Proof. See [59], Proposition 1.1.4 for more details. * * _|_| Remark. 3.4. If P is the maximal perfect normal subgroup of ss1(X) , it is usua* *l to write X+ for X+P. Let us come back to the question of the correspondence between group theory and* * topology. If G is a group and P a perfect normal subgroup of G , it is indeed a very good idea to look at* * the space BG+Psince the 11 celebrated Kan-Thurston theorem asserts that every topological space is of that* * form (see [52], [61], [28] and [49]). Theorem 3.5. (Kan-Thurston) For every connected CW-complex X there exists a gro* *up GX and a map tX : BGX = K(GX ; 1) ! X which is natural with respect to X and has the fol* *lowing properties: (a) the homomorphism (tX )* : ss1(BGX ) ~=GX ! ss1(X) induced by tX is surjecti* *ve, ~= (b) the map tX induces an isomorphism (tX )* : H*(BGX ; A) -! H*(X; A) for any * *local coefficient system A on X . This implies the following consequences. Corollary 3.6. For every connected CW-complex X , the kernel PX of (tX )* : GX!* *! ss1(X) is perfect. Proof. Let us look at the pull-back Y of the diagram Y ------! BGX ?? ? y etX ?ytX Xe ------! X ; in which Xe is the universal cover of X . Both horizontal maps have the same h* *omotopy fiber ss1(X) and tX induces an isomorphism on homology with any local coefficient system. Th* *erefore, the comparison theorem for the Serre spectral sequences of both horizontal maps implies that (* *etX)* : H1(Y ; Z) ! H1(Xe; Z) is an isomorphism and that H1(Y ; Z) vanishes since H1(Xe; Z) = 0 . The homotop* *y exact sequence of the fibration Y -! BGX -! Bss1(X) shows that ss1(Y ) ~=PX . Consequently, (PX )ab~= ss1(Y )ab~= H1(Y ; Z) = 0 , * *in other words, PX is a perfect group. * * _|_| Theorem 3.7. For every connected CW-complex X , there exists a group GX togeth* *er with a perfect normal subgroup PX such that one has a homotopy equivalence (BGX )+PX' X . Proof. For a connected CW-complex X , let us consider the group GX , the map tX* * : BGX ! X and the perfect group PX = ker((tX )* : GX!! ss1(X)) given by Theorem 3.5 and Corollar* *y 3.6. Then, consider the plus construction (BGX )+PXand apply Theorem 3.1 (c): there is a map t+X: (* *BGX )+PX! X which induces an isomorphism on ss1 and on all homology groups. The generalized White* *head theorem (see [48], Corollary 1.5, or [29], Proposition 4.15) finally implies that t+Xis a homotopy* * equivalence. _|_| Definition 3.8. (See [49], Sections 1 and 2.) A topogenic group is a pair (G; * *P) , where G is a group and P a perfect normal subgroup of G . In particular, a perfect group P can be * *viewed as a topogenic group because of the pair (P; P) . A morphism of topogenic groups f : (G; P) !* * (G0; P0) is a group homomorphism f : G ! G0 such that f(P) P0. An equivalence of topogenic groups* * is a morphism ~= f : (G; P) ! (G0; P0) inducing an isomorphism G=P -! G0=P0 and an isomorphism * *on homology f* : ~= H*(G; A) -! H*(G0; A) for all coefficients A . Consequently, if two topogenic g* *roups are equivalent, there is a map Bf+ : BG+P! BG0+P0between the corresponding CW-complexes which induces* * an isomorphism 12 on the fundamental group and on all homology groups. Again, it follows from the* * generalized Whitehead theorem that Bf+ : BG+P! BG0+P0is a homotopy equivalence. Consequently, this establishes a very nice one-to-one correspondence between gr* *oup theory and topology (see [28], Section 11, and [49], Section 2): equivalence classes homotopy types of topogenic groups! of CW-complexes (G; P) -! BG+P (GX ; PX ) - X Remark 3.9. By Theorem 3.1 the space BG+Passociated with the topogenic group (G* *; P) satisfies the following properties: (a) ss1(BG+P) ~=G=P , (b) Hi(BG+P; A) ~=Hi(G; A) for any coefficients A . Example 3.10. The perfect groups correspond to the simply connected CW-complexe* *s, because for any perfect group P , the space BP+Passociated with the topogenic group (P; P) has * *trivial fundamental group ss1BP+P~=P=P = 0 . Example 3.11. Let 1 = lim-!nn be the infinite symmetric group and A1 = lim-!* *nAn the infinite alternating group which is perfect. The Barratt-Priddy theorem [26] asserts th* *at the topogenic group (1 ; A1 ) corresponds to (B1 ) +A1which is homotopy equivalent to the connected* * component of the space Q0S0 = lim-!nnSn whose homotopy groups are the stable homotopy groups of * *spheres ssi(Q0S0) = lim-!nssi(nSn) = lim-!nssi+n(Sn) , i 0 . In order to define a suitable generalization of the functor K1(-) and K2(-) for* * rings, let us consider again the infinite general linear group GL(R) with coefficients in a ring R and* * its perfect normal subgroup generated by elementary matrices E(R) : we get the topogenic group (GL(R); E(R)* *) . The higher algebraic K-theory of R is the study of the corresponding topological space BGL(R)+E(R)(f* *or simplicity, we shall write BGL(R)+ for BGL(R)+E(R)according to Remark 3.4). Definition 3.12. (Quillen) (See [70].) For any ring R and any positive integer* * i , the i-th algebraic K-theory group of R is Ki(R) = ssi(BGL(R)+) : Let us check that this definition extends the definition of K1(R) and K2(R) giv* *en in Section 2. Remark 3.9 (a) shows that ss1(BGL(R)+) ~=GL(R)=E(R) and this group is exactly K1(R) ac* *cording to Definition 2.6. Notice also that Lemma 3.3 shows that the universal cover of BGL(R)+ is the spa* *ce BE(R)+ associated with the topogenic group (E(R); E(R)) . Therefore, we get the following result. Theorem 3.13. For any ring R the space BE(R)+ is simply connected and for all i* *ntegers i 2 , Ki(R) ~=ssi(BE(R))+ : In particular, the group K2(R) given by Definition 3.12 coincides with the grou* *p ss2(BE(R)+) which is isomorphic to H2(BE(R)+; Z) ~=H2(E(R); Z) (see Remark 3.9 (b)) because of the H* *urewicz theorem. Corollary 2.25 then implies that it is isomorphic to K2(R) as defined in Defini* *tion 2.23. 13 Of course, a ring homomorphism f : R ! R0 induces a group homomorphism GL(R) ! * *GL(R0) whose re- striction to E(R) sends E(R) into E(R0) . Thus, Corollary 3.2 implies the exist* *ence of a map BGL(R)+ ! BGL(R0)+ and consequently of a homomorphism of abelian groups f* : Ki(R) ! Ki(R* *0) for all i 1 . Moreover, one can check that Kn(-) is a covariant functor from the category of * *rings to the category of abelian groups. As we just observed, one can express the first two K-groups homologically (see * *Corollaries 2.7 and 2.25): K1(R) ~=H1(GL(R); Z) ; K2(R) ~=H2(E(R); Z) : In the same way, we can prove the following result. Theorem 3.14. Let St(R) be the infinite Steinberg group over a ring R . Then (a) the space BSt(R)+ is 2-connected, (b) Ki(R) ~=ssi(BSt(R)+) for all i 3 , (c) K3(R) ~=H3(St(R); Z) . Proof. Let us consider the universal central extension 0 -! K2(R) -! St(R) -'!E(R) -! 1 and the associated fibration of classifying spaces BK2(R) -! BSt(R) B'-!BE(R) : Since E(R) and St(R) are perfect, one can perform the + construction to both sp* *aces BSt(R) and BE(R) . If one denotes by F the homotopy fiber of the induced map + BSt(R)+ B'-!BE(R)+ ; one has the following commutative diagram where both rows are fibrations: BK2(R) ------! BSt(R) ---B'---! BE(R) ?? ? ? yf ?y+ ?y+ + F ------! BSt(R)+ --B'----!BE(R)+ : Since K2(R) is the center of St(R) by Theorem 2.22, the action of ss1(BE(R)) on* * the homology of BK2(R) is trivial. The same holds for the second fibration since BE(R)+ is simp* *ly connected. The two right vertical arrows induce isomorphisms on integral homology by Theorem 3.1. * *Therefore, the comparison theorem for spectral sequences implies that f* : H*(BK2(R); Z) ! H*(F; Z) is an* * isomorphism. Since St(R) is perfect, the space BSt(R)+ is also simply connected. On the other ha* *nd, it is known that H2(BSt(R)+; Z) ~=H2(St(R); Z) = 0 according to [54]. Consequently, the Hurewicz* * theorem shows that BSt(R)+ is actually 2-connected. Now, let us look at the homotopy exact sequence + ss2(BSt(R)+)_____-z____"-! ss2(BE(R)+)_____-z____"B'*-!ss* *1(F) -! 0 = 0 ~=K2(R) of the fibration F -! BSt(R)+ -! BE(R)+ : 14 Since ss1(F) ~=K2(R) , it is an abelian group. Consequently, the Hurewicz homo* *morphism ss1(F) ! H1(F; Z) is an isomorphism. Then, consider the commutative diagram ss1(BK2(R)) ---f*---! ss1(F) ?? ? y ?y~= H1(BK2(R); Z) ---f*---!~=H1(F; Z) ; where the vertical arrows are Hurewicz homomorphisms. The left vertical arrow i* *s an isomorphism since BK2(R) is an Eilenberg-MacLane space K(K2(R); 1) with K2(R) abelian. Thus, f* :* * ss1(BK2(R)) ! ss1(F) is an isomorphism and f : BK2(R) ! F is a homotopy equivalence because * *of the generalized Whitehead theorem (see [48], Corollary 1.5, or [29], Proposition 4.15). In oth* *er words, one obtains the following fibration (which can also be deduced from a more general topological * *argument, see [29], Theorem 6.4, [59], Theoreme 1.3.5, or [96], Lemma 3.1): BK2(R) -! BSt(R)+ -! BE(R)+ : The homotopy exact sequence of that fibration shows that ~= + ssi(BSt(R)+) -! ssi(BE(R) ) ~=Ki(R) B'+* for all i 3 and that BSt(R)+ is the 2-connected cover of BGL(R)+ . Finally, * *it follows from the Hurewicz theorem that K3(R) ~=ss3(BSt(R)+) ~=H3(BSt(R)+; Z) ~=H3(St(R); Z) : * *_|_| Remark 3.15. This homological interpretation of the groups Ki(R) for i = 1 , 2 * *, 3 , suggests the following generalization. Let us denote by BGL(R)+(m) the m-connected cover of the space * *BGL(R)+ for m 0 ; more precisely, BGL(R)+(m) is m-connected and there is a map BGL(R)+(m) ! BGL(R* *)+ inducing an isomorphism on ssi for i m + 1 . For instance, BGL(R)+(1) = BE(R)+ and BGL(R)+* *(2) = BSt(R)+ . According to the Kan-Thurston theorem (see Theorem 3.5), there exists a perfect* * group Gm (R) for each positive integer m such that BGm (R)+ ' BGL(R)+(m) . Consequently, Ki(R) ~=ssi(BGL(R)+(m)) ~=ssi(BGm (R)+) for i m + 1 and Km+1(R) ~=ssm+1(BGm (R)+) ~=Hm+1(BGm (R)+; Z) ~=Hm+1(Gm (R); Z) since BGm (R)+ is m-connected. Thus, there exists a list of groups G0(R) = GL(R* *) , G1(R) = E(R) , G2(R) = St(R) , G3(R) , G4(R) , ... whose homology represents the K-groups of R* * . Unfortunately, we do not have any explicit description of the groups Gm (R) for m 3 . Remark 3.16. D. Quillen also gave another equivalent definition of the higher K* *-groups. Let R be a ring and consider again the category P(R) of finitely generated projective R-modules* *. He constructed a new category QP(R) and its classifying space BQP(R) . Furthermore, he defined Ki(R) = ssi+1(BQP(R)) for i 0 . In fact, it turns out that the loop space BQP(R) of BQP(R) satisfies BQP(R) ' K0(R) x BGL(R)+ 15 for any ring R (see [72], Sections 1 and 2, or [74], Theorem 1, for the details* * of that construction). Of course, Quillen proved that both definitions of the K-groups coincide. In the present paper we want to concentrate our attention on the first definiti* *on of the K-groups (see Definition 3.12). The higher algebraic K-theory of a ring R is really the study* * of the space BGL(R)+ whose homotopy type is determined by its homotopy groups Ki(R) and by its Postn* *ikov k-invariants (see Section 7). The space BGL(R)+ has actually many other interesting properties. T* *he remainder of the paper is devoted to the investigation of some of them. 4. The product structure in algebraic K-theory and the K-theory spectrum The goal of this section is to show that the algebraic K-theory space BGL(R)+ o* *f any ring R has actually a very rich structure. Let us start by considering a ring R and the homomorphism : GL(R) x GL(R) -! GL(R) given by 8 < ffkl, if i = 2k - 1 and j = 2l - 1, (ff fi)ij= : fikl, if i = 2k and j = 2l, 0 , otherwise, for ff , fi 2 GL(R) . Since there is a homotopy equivalence BGL(R)+ x BGL(R)+ '* * B(GL(R) x GL(R))+ (see [59], Proposition 1.1.4), we can define the map + : BGL(R)+ x BGL(R)+ ' B(GL(R) x GL(R))+ -! BGL(R)+ which endows BGL(R)+ with the following structure. Proposition 4.1. For any ring R , the space BGL(R)+ , together with the map , * *is a commutative H-group. Proof. See [59], Theoreme 1.2.6. * * _|_| Now, let R and R0 be two rings and let us denote by R R0 the tensor product R * *Z R0 over Z . The tensor product of matrices induces a homomorphism GLm (R) x GLn(R0) -! GLmn(R R0) and a map 0 ejR;Rm;n: BGLm (R)+ x BGLn(R0)+ -! BGLmn(R R0)+ : By composing this map with the map induced by the upper left inclusion GLmn(R * *R0) ,! GL(R R0) we get a map 0 jR;Rm;n: BGLm (R)+ x BGLn(R0)+ -! BGL(R R0)+ : Then, let us define the map 0 + 0 + 0 + flR;Rm;n: BGLm (R) x BGLn(R ) -! BGL(R R ) by flR;R0m;n(x; y) = jR;R0m;n(x; y)-jR;R0m;n(x0; y)-jR;R0m;n(x; y0) , where x0 * *and y0 are the base points of BGLm (R)+ and BGLn(R0)+ respectively, and where " - " is the subtraction in the sense of * *the H-space structure of the space BGL(R R0)+ . Since the maps flR;R0m;nare compatible (up to homotopy)* * with the stabilizations 16 0 iR;Rm;n: BGLm (R)+ x BGLn(R0)+ ! BGLm+1(R)+ x BGLn+1(R0)+ induced by upper left* * inclusions, i.e., 0 0 flR;R0m;n' flR;Rm+1;n+1iR;Rm;n, (see [59], Lemme 2.1.3), we get a map 0 + 0 + 0 + flR;R : BGL(R) x BGL(R ) -! BGL(R R ) ; which is unique up to weak homotopy (see [59], Lemme 2.1.6 and Remarque 2.1.9).* * By definition, this map flR;R0is homotopic to the trivial map on the wedge BGL(R)+ _BGL(R0)+ . Conseque* *ntly, it finally induces a map 0 + 0 + 0 + bflR;R: BGL(R) ^ BGL(R ) -! BGL(R R ) : It turns out that this map bflR;R0is natural in R and R0, bilinear, associative* * and commutative, up to weak homotopy (see [59], Proposition 2.1.8). It enables us to give the following def* *inition (see [59], Definition 2.1.10; an alternative definition can be found in Chapter 13 of [29]). Definition 4.2. (Loday) For all rings R and R0, and for all integers i , j 1 ,* * the product map ? : Ki(R) x Kj(R0) = ssi(BGL(R)+) x ssj(BGL(R0)+) -! ssi+j(BGL(R R0)+) = K* *i+j(R R0) ; is defined as follows: if x 2 Ki(R) and y 2 Kj(R0) are represented by ff : Si! * *BGL(R)+ and fi : Sj ! BGL(R0)+ respectively, then R;R0 x ? y = [Si+j' Si^ Sj ff^fi-!BGL(R)+ ^ BGL(R0)+ bfl-!BGL(R R0)+] : One can then immediately deduce the following properties (see [59], Theoreme 2.* *1.11). Proposition 4.3. The product map ? : Ki(R)xKj(R0) ! Ki+j(R R0) is natural in R * *and R0, bilinear and associative for all i , j 1 . Remark 4.4. Because of that proposition, we can consider the above product on * *the tensor product Ki(R) Kj(R0) and we shall also denote it by the symbol ? : ? : Ki(R) Kj(R0) -! Ki+j(R R0) : Now, let us look at the special case where R0= R . If R is a commutative ring, * *the ring homomorphism r : R R ! R given by r(a b) = ab induces a ring structure on K*(R) . Definition 4.5. If R is a commutative ring, then there is a product map (also d* *enoted by ? ) ? : Ki(R) Kj(R) -?!Ki+j(R R) r*-!Ki+j(R) for all i , j 1 . This product satisfies: Proposition 4.6. If R is a commutative ring, then for all x 2 Ki(R) and y 2 Kj(* *R) with i; j 1 , one has x ? y = (-1)ijy ? x . Proof. Let again x 2 Ki(R) and y 2 Kj(R) be represented by ff : Si ! BGL(R)+ a* *nd fi : Sj ! BGL(R)+ respectively. Let t : R R ! R R denote the homomorphism given by t(a * * b) = b a and 17 s : Si^ Sj ! Sj^ Si the homeomorphism which exchanges the factors. Since R is c* *ommutative, r t = r and we get the commutative diagram R;Rff^fi r+ Si^ Sj bfl------!BGL(R R)+ ------! BGL(R)+ ?? ? ? ys ?yt+ ?yid R;Rfi^ff r+ Sj^ Si bfl------!BGL(R R)+ ------! BGL(R)+ which provides the result since the homotopy class of s is (-1)ij2 ssi+j(Si+j) * *~=Z . _|_| The remainder of this section is devoted to further investigation of the H-spac* *e structure of the space BGL(R)+ (see [59], Sections 1.4 and 2.3). Let us first consider the ring of int* *egers R = Z . Its cone CZ is the set of all infinite matrices with integral coefficients having only a finit* *e number of non-trivial elements on each row and on each column. This set turns out to be a ring by the usual ad* *dition and multiplication of matrices. Let JZ be the ideal of CZ which consists of all matrices having on* *ly finitely many non-trivial coefficients. Finally, let us define the suspension of Z to be the quotient rin* *g Z = CZ=JZ . Definition 4.7. For any ring R , the suspension of R is the ring R = Z Z R : Let 0 1 0 0 0 0 . . . BB1 0 0 0 . .C.C o = BBB01000100 ......CC2 Z : @ . . . . . .C.A . . . . . . . This element is invertible since oot = 1 in Z and consequently o 2 GL1(Z) . Let* * [P] be a generator of the group K0(R) , where P is a finitely generated projective R-module. There* * is an R-module Q such that P Q ~=Rn for some n and an R-module homomorphism Rn ! Rn which is the ide* *ntity on P and trivial on Q . Let us call p the n x n matrix with coefficients in R correspond* *ing to that homomorphism. By using the tensor product of matrices, one can construct the element o p + 1* * (1 - p) , which is an invertible n x n matrix with coefficients in R , in other words, which belongs * *to GL(R) . This produces a homomorphism : K0(R) ! K1(R) which sends [P] to the class of o p + 1 (1 - p) in K1(R) = GL(R)=E(R) . Proposition 4.8. The homomorphism : K0(R) ! K1(R) is an isomomorphism. This fact is proved in [53] and can be generalized. Let oe : Z ! GL1(Z) be the * *group homomorphism given by oe(1) = o . It induces a map oe+ : S1 ' BZ+ ! BGL(Z)+ and we write e"R* *for the composition +^id bflZ;R e"R: S1^ BGL(R)+ oe-!BGL(Z)+ ^ BGL(R)+ -! BGL(R)+ : For any generator [P] of K0(R) , let us choose a representative iP : S1 ! BGL(R* *)+ of the element ([P]) 2 K1(R) = ss1(BGL(R)+) . This defines a map "0R: S1^ (K0(R) x BGL(R)+) -! K0(R) x BGL(R)+ 18 given by "0R(t ^ ([P]; x)) = (0; iP(t) + e"R(t ^ x)) . Its adjoint is "R : K0(R) x BGL(R)+ -! (K0(R) x BGL(R)+) : The point is that the homotopy type of the space BGL(R)+ depends very strongly * *on the homotopy type of BGL(R)+ because of the following result. Theorem 4.9. The map "R is a natural homotopy equivalence: "R : K0(R) x BGL(R)+ -'!(K0(R) x BGL(R)+) : Proof. See [59], Theoreme 1.4.9 and Theoreme 2.3.5; see also [96], Section 3. * * _|_| Theorem 4.9 immediately implies the following consequence. Corollary 4.10. For any ring R and for any integer i 0 , there is an isomorphi* *sm Ki(R) ~=Ki+1(R) : Proof. By definition, Ki(R) = ssi(K0(R) x BGL(R)+) ~=ssi+1(K0(R) x BGL(R)+) ~=ssi+1(BGL(R)+) = Ki+* *1(R) for any integer i 0 . * * _|_| If one takes the 0-connected cover of both sides of the equivalence provided by* * Theorem 4.9 and applies Theorem 3.13, one gets: Corollary 4.11. There is a natural homotopy equivalence BGL(R)+ ' BE(R)+ : Remark 4.12. Of course, Theorem 4.9 shows that for any ring R , the space K0(R)* *xBGL(R)+ is an infinite loop space since K0(R) x BGL(R)+ ' (K0(R) x BGL(R)+) ' 2(K0(2R) x BGL(2R)+) ' .* * ... This enables us to define an -spectrum whose 0-th space is the space K0(R) x BG* *L(R)+ . Definition 4.13. For any ring R , the K-theory spectrum of R is the -spectrum K* *R whose n-th space is (KR)n = K0(nR) x BGL(nR)+ for all n 0 . We shall also use the 0-connected cover XR of the K-theory spectrum KR of a rin* *g R . Definition 4.14. For any ring R , the 0-connected K-theory spectrum of R is the* * -spectrum XR whose n-th space is (XR)n = BGL(nR)+(n) for all n 0 . Here, X(n) is written for the * *n-th connected cover of a CW-complex X , i.e., the homotopy fiber of the n-th Postnikov section X ! * *X[n] of X (see Section 7): this means that X(n) is n-connected and that ssi(X(n)) ~=ssi(X) for all i * *n . Of course, since these spectra are -spectra, their homotopy groups are ssi(KR) = lim-!ssi+n((KR)n) = lim-!ssi+n(K0(nR) x BGL(nR)+) ~=lim-!Ki+* *n(nR) n n n for any i 2 Z ; in particular, they are in general non-trivial if i < 0 . On th* *e other hand, for XR , ssi(XR) = lim-!ssi+n((XR)n) = lim-!ssi+n(BGL(nR)+(n)) n n 19 and we may conclude that ssi(XR) = 0 if i 0 since BGL(nR)+(n) is n-connected a* *nd that ssi(XR) ~=lim-!ssi+n(BGL(nR)+) ~=lim-!Ki+n(nR) n n if i 1 . Therefore, Corollary 4.10 implies the following result for i 0 . Theorem 4.15. (a) For any integer i 0 , Ki(R) ~=ssi(KR) . (b) For any integer i 1 , Ki(R) ~=ssi(XR) . Remark 4.16. There are many other constructions of the K-theory spectrum of a r* *ing R : see for instance [29], Chapter 11, [45], or [96], Section 3. Remark 4.17. Since the K-groups of a ring R in positive dimensions are the homo* *topy groups of its 0-connected K-theory spectrum XR (or of KR ), they are very strongly related to* * the homology groups of XR via the stable Hurewicz homomorphism, as we shall see in Section 6. Therefor* *e, it would be extremely useful to be able to compute Hi(XR; Z) = lim-!Hi+n((XR)n; Z) = lim-!Hi+n(BGL(nR)+(n); Z) : n n In a recent paper [63] which generalizes MacLane's Q-construction for computing* * the stable homology of Eilenberg-Maclane spaces [60], R. McCarthy obtains an explicit chain complex* * whose homology is the homology of XR . This promising idea should provide more information on Hi(XR; * *Z) and consequently on the algebraic K-groups Ki(R) . Let us conclude this section by explaining that the product structure in the K-* *theory of rings may also be expressed in terms of K-theory spectra. Definition 4.18. Let us consider two rings R and R0, together with their associ* *ated 0-connected spectra XR and XR0, and let S denote the sphere spectrum. The external product ^ : ssi(XR) ssj(XR0) -! ssi+j(XR ^ XR0) is defined as follows. If x 2 ssi(XR) and y 2 ssj(XR0) are represented by maps * *of spectra ff : S ! XR of degree i and fi : S ! XR0 of degree j respectively, x ^ y is then the class in * *ssi+j(XR ^ XR0) represented by the map ff ^ fi : S ' S ^ S ! XR ^ XR0 of degree i + j (see [94], p.270). Definition 4.19. The map bflR;R0: BGL(R)+^BGL(R0)+ -! BGL(RR0)+ which was the k* *ey ingredient in Definition 4.2 extends of course to a map nR;mR0 n + m 0 + n+m 0 + bfl : BGL( R) ^ BGL( R ) -! BGL( (R R )) for all n and m and consequently to a pairing of spectra _flR;R0: X R ^ XR0-! XRR0 which we call the Loday pairing (see [59], Proposition 2.4.2). Therefore, the definition of the K-theoretical product introduced in Definition* * 4.2 and Remark 4.4 can be formulated as follows. 20 Corollary 4.20. For all rings R and R0 and for all positive integers i and j , * *the K-theoretical product is given by _flR;R0)* ? : Ki(R) Kj(R0) ~=ssi(XR) ssj(XR0) -^!ssi+j(XR ^ XR0) (-! ssi+j(XRR0 ) ~* *=Ki+j(R R0) : Moreover, if R is commutative, the ring structure of K*(R) is given by _flR;R)* r ? : Ki(R)Kj(R) ~=ssi(XR)ssj(XR) -^!ssi+j(XR ^XR) (-! ssi+j(XRR ) ~=Ki+j(RR) -** *!Ki+j(R) for all i , j 1 . 5. The algebraic K-theory of finite fields When D. Quillen introduced the higher algebraic K-groups, one of his great achi* *evements was to completely compute them for finite fields (see [71]). Let p be a prime, q a power of p and* * let Fq denote the field with q elements. Since Fq is a field, it is known by Theorem 1.2 that K0(Fq) ~=Z : In order to calculate Ki(Fq) for any positive integer i , Quillen's brilliant i* *dea was to construct a topological model for the space BGL(Fq)+ using well known spaces. He considered the classif* *ying space BU of the infinite unitary group U and the Adams operation q: BU ! BU . Remember that for* * i 1 ssi(BU) = 0 if i is odd and ssi(BU) ~=Z if i is even (by Bott periodicity, see [35] or Chap* *ter 10 of [51]), and that the homomorphism q*: ss2j(BU) ! ss2j(BU) induced by q is multiplication by qj (see * *[1], Corollary 5.2). Definition 5.1. For any integer q 2 , let Fq be the pull-back of the diagram Fq ---'---! BU ?? ? y ?y(q;id) BU[0;1]------! BU x BU ; where BU[0;1]is the path space of BU and the map sending a path in BU to its e* *ndpoints. A point of Fq is a pair (x; u) , where x is a point of BU and u a path in BU joining q(* *x) to x . In other words, Fq is the homotopy theoretical fixpoint set of q. According to Lemma 1 o* *f [71], it turns out that if d : BU x BU ! BU is the map defined by d(x; y) = x - y , then Fq is the* * homotopy fiber of the composition q;id) d q- 1 : BU (-! BU x BU -! BU : Proposition 5.2. For any integer q 2 , the space Fq is simple and its homotopy* * groups are ae0 , if i is an even integer 2, ssi(Fq) ~= j Z=(q - 1) , if i is an odd integer of the form i = 2j - 1 w* *ith j 1. Proof. Let us consider the fibration q-1 Fq-'! BU -! BU : 21 Since the action of ss1(Fq) on the higher homotopy groups ssi(Fq) comes from th* *e action of ss1(BU) on ssi(Fq) , it is trivial because BU is simply connected. The calculation of ssi(* *Fq) directly follows from the homotopy exact sequence of the above fibration j-1 . .-.! ss2j+1(BU)-! ss2j(Fq) -! ss2j(BU)q-!ss2j(BU)-! ss2j-1(Fq) -! ss2j-1(B* *U)-! . . . ____-z___" ___-z__" ___-z__" ____-z__* *_" =0 ~=Z ~=Z =0 for any positive integer j . * * _|_| From now on, let p be a prime, q a power of p and fix a prime l such that l 6= * *p . Quillen's main argument is based on the calculation of the cohomology of the space Fq and of the infini* *te general linear group GL(Fq) . Let us start by defining some classes in the cohomology of Fq. It is w* *ell known that H*(BU; Z) ~=Z[bc1; bc2; bc3; : :]: and H*(BU; Z=l) ~=Z=l[_c1; _c2; _c3; : :]:; where the bci's and the _ci's are the integral universal Chern classes, respect* *ively the mod l universal Chern classes, of degree 2i (see [66], Chapter 14). Definition 5.3. For any positive integer i , the i-th integral Chern class of F* *q is eci= '*(bci) 2 H2i(Fq; Z) and the i-th mod l Chern class of Fq is ci= '*(_ci) 2 H2i(Fq; Z=l) ; where '* : H*(BU; A) ! H*(Fq; A) is the homomorphism induced by ' , for A = Z a* *nd A = Z=l respectively. The diagram occuring in Definition 5.1 induces the following commutative diagra* *m for any abelian group A in which the rows are exact sequences of pairs: * H2i-1(Fq; A) -ffi-! H2i(BU; Fq; A) -fl-! H2i(BU; A) -'-! H2i(Fq; A) x? x x x ? ??i ??(q;id)* ?? 0 fl0 * H2i-1(BU[0;1];-A)ffi-!H2i(BU x BU; BU[0;1];-A)-!H2i(BU x BU;-A)-!H2i(BU[0;1]* *; A) : In this diagram, fl0 and ffi are injective because H2i-1(BU[0;1]; A) = H2i-1(BU* *; A) = 0 . Let us first take A = Z . Since : BU[0;1]! BU x BU is homotopy equivalent to the diagonal map BU* * ! BU x BU sending a point x to (x; x) , the induced homomorphism * : H2i(BU xBU; Z) ! H2i* *(BU[0;1]; Z) satisfies *(bci 1 - 1 bci) = 0 . Consequently, there is a unique element z 2 H2i(BU x BU* *; BU[0;1]; Z) such that fl0(z) = bci1-1bci. Thus, the commutativity of the diagram shows that i(z) 2 H2* *i(BU; Fq; Z) has the property that fl i(z) = (qi-1)bci2 H2i(BU; Z) . Now, consider both coefficients* * A = Z and A = Z=(qi-1) , and the diagram 0 --! H2i-1(Fq; Z) --ffi! H2i(BU; Fq; Z) --fl! H2i(BU; Z) ?? ? ? y red(qi-1) ?yred(qi-1) ?yred(qi-1) 0 --! H2i-1(Fq; Z=(qi- 1))-ffi-!H2i(BU; Fq; Z=(qi- 1))fl--!H2i(BU; Z=(qi- * *1)) ; 22 where the vertical homomorphisms are induced by the reduction mod (qi- 1) . It * *follows from the commu- tativity of the diagram that fl red(qi-1)i(z) = red(qi-1)((qi- 1)bci) = 0 . Definition 5.4. For any positive integer i , there is a unique element eei2 H2i-1(Fq; Z=(qi- 1)) such that ffi(eei) = red(qi-1)i(z) . This element is related to the integral Ch* *ern class eci2 H2i(Fq; Z) by the formula fi(qi-1)(eei) = eci; where fi(qi-1)denotes the Bockstein homomorphism H2i-1(Fq; Z=(qi- 1)) ! H2i(Fq;* * Z) (see Lemmas 3 and 5 in [71]). Definition 5.5. Let r be the smallest positive integer such that qr 1 mod l . * *Then, we define for any integer j 1 ejr2 H2jr-1(Fq; Z=l) as the image of eejrunder the homomorphism H2jr-1(Fq; Z=(qjr- 1)) ! H2jr-1(Fq; * *Z=l) induced by the obvious surjection Z=(qjr- 1)!! Z=l . By using the Eilenberg-Moore spectral sequence of the fibration q-1 Fq-'! BU -! BU ; D. Quillen was able to calculate the cohomology of Fq with coefficients in Z=l . Theorem 5.6. (a) The monomials cff1rcff22rcff33r.e.f.i1refi22refi33r.,.w.ith ffj 0 and fij * *= 0 or1 , form an additive basis for H*(Fq; Z=l) . (b) If l is an odd prime or if l = 2 and q 1 mod 4 , then e2jr= 0 for all j 1* * and there is an algebra isomorphism H*(Fq; Z=l) ~=Z=l[cr; c2r; c3r; : :]: Z=l(er; e2r; e3r; : :* *):: (c) If l = 2 and q 3 mod 4 , one has r = 1 and the relations j-1X e2j= c2j-1+ ckc2j-k-1; k=1 and there is an algebra isomorphism H*(Fq; Z=2) ~=Z=2[c2; c4; c6; : :;:e1; e2; e3; : :]:: Proof. See [71], Theorem 1 and [44], Section IV.8. * * _|_| The next ingredient in Quillen's argument is the notion of the Brauer lift (see* * [71], Section 7). Let G be a finite group and ae : G ! GLn(Fq) a_representation of G over the field_Fq with * *q elements. Let us denote by _aethe representation _ae= aeFqFqof G over the algebraic closure Fq of Fq. W* *e can look at the complex P * * __* valued function on G defined by O_ae(g) = (k(g)) for g 2 G , where is an emb* *edding Fq ,! C* and {k(g)} is the set of eigenvalues of _ae(g) . It turns out that O_aeis the chara* *cter of a unique virtual complex representation eaeof G ; therefore, O_aebelongs to the complex representation r* *ing R(G) = RC(G) of G and we get a homomorphism RFq(G) ! R(G) which maps the class of the character o* *f ae to O_ae. In fact, eaeis stable under the Adams operation q (see [71], section 7) and the pr* *evious homomorphism is 23 q actually RFq(G) ! R(G) . If we compose it with the classifying map R(G) ! [BG* *; BU] sending a complex representation to the corresponding homotopy class of maps between clas* *sifying spaces, we obtain the homomorphism q o : RFq(G) ! [BG; BU] : q-1 On the other hand, observe again the fibration Fq-'! BU -! BU and remember tha* *t a point of Fq is a pair (x; u) , where x is a point of BU and u a path joining q(x) to x : th* *is implies that for any Y , a map Y ! Fq can be identified with a pair consisting of a map f : Y ! BU toget* *her with a homotopy joining qf to f . Consequently, ' induces a homomorphism q '* : [Y; Fq] -! [Y; BU] which is clearly surjective. By looking at the fibration BU ' U -! Fq-'! BU obtained by looping the base space of the above fibration, one gets that '* is * *an isomorphism if [Y; U] = 0 . Therefore, if [BG; U] = 0 , the above homomorphism o can be viewed as a homomor* *phism o : RFq(G) ! [BG; Fq] : This is the case for G = GLn(Fq) and for the direct limit G = GL(Fq) = lim-!nGL* *n(Fq) according to Lemma 14 of [71]. Definition 5.7. Let G = GLn(Fq) and ae = id: GLn(Fq) ! GLn(Fq) . The Brauer lif* *t is the homotopy class of maps bn = o(id) 2 [BGLn(Fq); Fq] . By passing to the direct limit GL(F* *q) = lim-!nGLn(Fq) , one obtains a homotopy class of maps b = o(id) 2 [BGL(Fq); Fq] : For simplicity, we shall also denote by b 2 [BGL(Fq); BU] the composition of th* *is last homotopy class of maps with the inclusion ' : Fq! BU and call it the Brauer lift. This enables Quillen to prove his main result. Theorem 5.8. (Quillen) For any prime power q , there is a homotopy equivalence BGL(Fq)+ ' Fq: Proof. (See [71], Theorems 2, 3, 4, 5, 6 and 7 for the details.) Let p be a pri* *me, q a power of p and r be as in Definition 5.5. The argument is based on the investigation of the map b+ : BGL(Fq)+ -! Fq induced by the the Brauer lift b : BGL(Fq) -! Fq (notice that (Fq)+ ' Fq since * *ss1(Fq) is abelian by Proposition 5.2 and contains therefore no non-trivial perfect normal* * subgroup). Because of Theorem 5.6, one can also compute the modl homology of Fq for any prime l 6= p* * . On the other hand, using techniques from homology theory of finite groups, it is possible to* * calculate H*(GLn(Fq); Z=l) for all positive integers n and consequently H*(GL(Fq); Z=l) ~=lim-!nH*(GLn(Fq)* *; Z=l) , and to prove that the homomorphism (b+)* : H*(BGL(Fq)+; Z=l) ~=H*(GL(Fq); Z=l) -! H*(Fq; Z=l) ind* *uced by b is an isomorphism. The next thing to do is to prove the vanishing of Hi(GL(Fq); Z=p) * *~=Hi(BGL(Fq)+; Z=p) for all i 1 . Then, apply the generalized Whitehead theorem (see [48], Corolla* *ry 1.5, or [29], Proposition 4.15) to the map b+ : BGL(Fq)+ -! Fq: since both spaces are simple according to* * Propositions 4.1 and 5.2, we can conclude that b+ is a homotopy equivalence if we can show that b in* *duces an isomorphism ~= q (b+)* : H*(BGL(Fq)+; Z) -! H*(F ; Z) : 24 This holds if b+ induces an isomorphism on homology with coefficients in Q , in* * Z=p and in Z=l for all primes l 6= p . This is already done for coefficients in Z=l . It is easy to ch* *eck that Hi(BGL(Fq)+; Q) ~= Hi(GL(Fq); Q) = 0 for all i 1 because Hi(GL(Fq); Q) ~=lim-!nHi(GLn(Fq); Q) = 0* * since GLn(Fq) is a finite group. On the other hand, we know from Proposition 5.2 that the homotopy* * groups of Fq are torsion groups which are p-torsion free. Thus, by Serre class theory (see [82], Chapitr* *e I), all integral homology groups of Fq are also torsion groups which are p-torsion free: in other words, * *Hi(Fq; Q) = 0 and Hi(Fq; Z=p) = 0 for all i 1 . Thus, b+ : BGL(Fq)+ ! Fq is a homotopy equivalen* *ce and we get the statement of the theorem. * * _|_| This result is important because it provides a convenient topological model Fq * *for the algebraic K-theory space BGL(Fq)+ . In particular, an immediate consequence of it is the calculati* *on of the algebraic K-groups of all finite fields: Proposition 5.2 and Theorem 5.8 imply the following resul* *t (see [71], Theorem 8). Corollary 5.9. For any prime power q , the algebraic K-theory of the finite fie* *ld Fq is given by K2i(Fq) = 0 and K2i-1(Fq) ~=Z=(qi- 1) for all integers i 1 . This result was the first determination of K-groups and initiated in some sense* * the research in the algebraic K-theory of rings. 6. The Hurewicz homomorphism in algebraic K-theory The computation of the algebraic K-groups of finite fields was the first impres* *sive K-theoretical result. It turns out that it is actually difficult to perform many other computations. How* *ever, this is not a surprise because the algebraic K-groups are homotopy groups and it is never easy to comp* *ute homotopy groups! On the other hand, there are many sophisticated techniques for the computation * *of the homology of groups. Notice for instance that Quillen's result on the K-groups of finite fields is a* *ctually based on homological calculations. Therefore, it is useful to investigate the relationships between * *the algebraic K-theory of a ring R and the homology of its infinite linear groups GL(R) , E(R) , or of its infin* *ite Steinberg group St(R) . They are exhibited by the Hurewicz homomorphisms hi: Ki(R) = ssi(BGL(R)+) -! Hi(BGL(R)+; Z) ~=Hi(GL(R); Z), fori 1 ; hi: Ki(R) ~=ssi(BE(R)+) -! Hi(BE(R)+; Z) ~=Hi(E(R); Z), fori 2 ; hi: Ki(R) ~=ssi(BSt(R)+) -! Hi(BSt(R)+; Z) ~=Hi(St(R); Z), fori 3* * : Of course, since BGL(R)+ , BE(R)+ and BSt(R)+ are connected, simply connected a* *nd 2-connected respectively (see Theorems 3.13 and 3.14), the classical Hurewicz theorem (see * *[100], Theorem IV.7.1) implies the following result. Theorem 6.1. For any ring R , (a) K1(R) ~=H1(GL(R); Z) , (b) K2(R) ~=H2(E(R); Z) and h3: K3(R) ! H3(E(R); Z) is surjective, (c) K3(R) ~=H3(St(R); Z) and h4: K4(R) ! H4(St(R); Z) is surjective. The general objective of this section is to approximate the size of the kernel * *and of the cokernel of hi in higher dimensions. We will proceed from different points of view (see also [7],* * [9], [12] and [14]). 25 Let us start by using stable homotopy theory (see also [14], Sections 1 and 2).* * For any spectrum X , the stable Hurewicz homomorphism is a homomorphism hi: ssi(X) -! Hi(X; Z) ; defined for all integers i , which fits into the long stable Whitehead exact se* *quence. This sequence can be defined as follows. Consider the sphere spectrum S . It is (-1)-connected wi* *th ss0(S) ~=Z and if we kill all its homotopy groups in positive dimensions, we get a map of spectra ff* *0 : S ! H(Z) inducing an isomorphism on ss0, where H(Z) is the Eilenberg-Maclane spectrum having all hom* *otopy groups trivial except ss0(H(Z)) ~=Z . The map ff0 is actually the 0-th Postnikov section of S * *(see Section 7). The stable Hurewicz homomorphism is the homomorphism hi: ssi(X) ~=ssi(X ^ S) -! ssi(X ^ H(Z)) ~=Hi(X; Z) induced by the map of spectra id^ ff0 : X ^ S ! X ^ H(Z) , where id is the iden* *tity : X ! X . Let us write S(0) for the homotopy fiber of ff0: in other words, S(0) is the 0-connect* *ed cover of S . By taking the smash product of X with the cofibration S(0) -fl0!S -ff0!S[0] = H(Z) , we o* *btain the cofibration of spectra X ^ S(0) id^fl0-!X ^ S ' X id^ff0-!X ^ H(Z) : Definition 6.2. The long stable Whitehead exact sequence of a spectrum X is the* * homotopy exact sequence of the above cofibration: hi i . .-.! ssi(X ^ S(0)) Oi-!ssi(X) -! Hi(X; Z) -! ssi-1(X ^ S(0)) -! .* * .:. Here i is any integer, Oi is induced by (id^ fl0) , hi is the stable Hurewicz h* *omomorphism and i is the connecting homomorphism. The groups ssi(X ^ S(0)) are usually denoted by i(* *X) : that definition coincides actually with the homotopy groups of the homotopy fiber of the Dold-T* *hom map (see [39]) and it was recently proved in [81], Corollary 3.9, that they are isomorphic to the gro* *ups introduced in the original paper [102] by J.H.C. Whitehead. Now, let us assume that the spectrum X is (b - 1)-connected for some integer b * *. The advantage of this approach is that one can compute the groups i(X) by using the Atiyah-Hirzebruch* * spectral sequence for S(0)-homology (see [2], Section III.7): E2s;t~=Hs(X; sst(S(0))) =) s+t(X) : Notice that E2s;t= 0 if s b - 1 or if t 0 . This reproves the Hurewicz theore* *m because i(X) = 0 for i b and consequently hi is an isomorphism for i b and an epimorphism for i = b + 1* * . Remark 6.3. For i = b + 1 , we get b+1(X) ~=E2b;1~=ssb(X) ss1(S) ~=ssb(X) Z=2 for any (b - 1)-connected spectrum X (this was already known by J.H.C. Whitehea* *d for any (b - 1)- connected spectrum or for any (b - 1)-connected space with b 3 , see [102], p.* *81, or [101]). Thus, the homomorphism Ob+1 is actually a homomorphism from ssb(X) ss1(S) to ssb+1(X* *) . Consider the 26 commutative diagram Hb(X; Z) H1(S(0); Z)-^-----!~=Hb+1(X ^ S(0); Z)) x? x ?~= ??~= ssb(X) ss1(S(0))---^---! ssb+1(X ^ S(0))= b+1(X) ? ? ~=?y(id)*(fl0)* ?yOb+1 ssb(X) ss1(S) ---^---! ssb+1(X ^ S)~= ssb+1(X) ; in which ^ is the external product (see Definition 4.18 or [94], p.270). The to* *p horizontal homomorphism is an isomorphism by K"unneth formula and the two top vertical arrows, which ar* *e Hurewicz homomor- phisms, are isomorphisms since X is (b-1)-connected, S(0) is 0-connected and X * *^S(0) is b -connected. Consequently, the external product in the middle of the diagram~is an isomorphi* *sm. The homomorphism (id)* (fl0)* is an isomorphism because (fl0)* : ss1(S(0)) -=!ss1(S) . Therefore* *, Ob+1 may be identified with the external product ssb(X) ss1(S) -^!ssb+1(X) . Thus, we proved the foll* *owing result. Proposition 6.4. For any (b - 1)-connected spectrum X , the sequence hb+2 b+2 ^ hb+1 . .-.! b+2(X) Ob+2-!ssb+2(X) -! Hb+2(X; Z) -! ssb(X) ss1(S) -! ssb+1(X) -! H* *b+1(X; Z) -! 0 is exact. Observe in particular that 2 (kerhb+1) = 0 and 2 (cokerhb+2) = 0 . Our first goal is to show that the spectral sequence E2s;t~=Hs(X; sst(S(0))) =) s+t(X) provides a generalization of that result for the exponent of all Gamma groups o* *f X . Definition 6.5. For any positive integer j , let ej be the exponent of the j-th* * homotopy group ssj(S) of the sphere spectrum S . For any positive integer i , let ei denote the product * *ei= e1e2e3. .e.i. Notice that a prime p divides ei if and only if p i+3_2according to Serre's theorem o* *n the stable homotopy groups of spheres (see [82], Section IV.6, Proposition 11). Now, if you look at the E2-term E2s;t~=Hs(X; sstS(0)) of the Atiyah-Hirzebruch * *spectral sequence for a (b - 1)-connected spectrum X , it is obvious that the product of the exponents * *of the groups E2s;t, for s + t = i with t 1 and s b , kills the Gamma group i(X) . Because etE2s;t= 0 for anyt 1 by Definition 6.5, since sst(S(0)) ~=sst(S) when t 1 , we conclude that the ex* *ponent of i(X) divides the product e1e2e3. .e.i-b. This immediately implies the following result which was* * also proved by a different argument in [81], Theorem 4.3, and in [12], Theorem 4.1. Theorem 6.6. Let X be a (b - 1)-connected spectrum. Then ei-bi(X) = 0 for all integers i b + 1 and the stable Hurewicz homomorphism hi: ssi(X) ! Hi(* *X; Z) satisfies: (a) ei-b(kerhi) = 0 for all integers i b + 1 , (b) ei-b-1(cokerhi) = 0 for all integers i b + 2 . 27 Of course, we want to apply this theorem to the K-theory spectrum. Let us consi* *der again the 0-connected K-theory spectrum XR of any ring R (see Definition 4.14) and let us kill its fi* *rst homotopy group: we get the 1-connected K-theory spectrum XR(1) . The above argument enables us to stud* *y the stable Hurewicz homomorphism hi: Ki(R) ~=ssi(XR(1)) ! Hi(XR(1); Z) which is an isomorphism if i* * = 2 . Theorem 6.6 holds here with b = 2 . Corollary 6.7. For any ring R , the stable Hurewicz homomorphism hi: Ki(R) ! Hi* *(XR(1); Z) satisfies: (a) ei-2(kerhi) = 0 for all integers i 3 , (b) ei-3(cokerhi) = 0 for all integers i 4 . In particular, the exponent of the kernel, respectively of the cokernel, of hi * *is only divisible by primes p i+1_2, respectively by primes p i_2. This can be formulated in another way. Definition 6.8. For any ring R , for any abelian group A and for any positive i* *nteger i , the i-th algebraic K-group of R with coefficients in A is the i-th homotopy group of BGL(R)+ or XR* * with coefficients in A (see [36] and [69]): Ki(R; A) = ssi(BGL(R)+; A) ~=ssi(XR; A) : In particular, if Z(p)denotes the ring of integers localized at p , then Ki(R; * *Z(p)) ~=Ki(R) Z(p)(see [36], Theorem 1.8, or [69], Proposition 1.4) and Corollary 6.7 shows (see also [12], * *Corollary 5.1): Corollary 6.9. For any ring R and any integer i 2 , Ki(R; Z(p)) ~=Hi(XR(1); Z(p)) for all prime numbers p i_2+ 1 . On the other hand, we can also deduce from the above considerations some inform* *ation on the unstable Hurewicz homomorphism hi: Ki(R) ~=ssi(BE(R)+) -! Hi(BE(R)+; Z) ~=Hi(E(R); Z) for i 2 . Since BE(R)+ is the 0-th space of the -spectrum XR(1) , we can look* * at the following commutative diagram for all integers i 2 : ~= Ki(R) ~=ssi(BE(R)+)------! ssi(XR(1)) ?? ? yhi ?yhi Hi(E(R); Z)H~=i(BE(R)+; Z)--oei----!Hi(XR(1); Z) ; where oei is the iterated homology suspension (see [100], Section VII.6 and Cha* *pter VIII). In order to state the next result, define ehi: Ki(R) ! Hi(E(R); Z)=(keroei) as the composit* *ion of hi : Ki(R) ! Hi(BE(R)+; Z) ~=Hi(E(R); Z) with the quotient map Hi(E(R); Z)!! Hi(E(R); Z)=(ke* *roei) . Corollary 6.10. For any ring R , the unstable Hurewicz homomorphism hi : Ki(R) * *! Hi(E(R); Z) satisfies: (a) ei-2(kerhi) = 0 for all integers i 3 , (b) ei-3(cokerehi) = 0 for all integers i 4 , (c) for all integers i 4 and for any integral homology class x 2 Hi(E(R); Z) ,* * there exists an element y in the image of hi: Ki(R) ! Hi(E(R); Z) and an element z in the kernel of t* *he iterated homology suspension oei: Hi(E(R); Z) ! Hi(XR(1); Z) such that ei-3x = y + z . 28 Proof. (See also [12], Corollary 5.2.) Because of the commutativity of the abov* *e diagram, Corollary 6.7 (a) implies Assertion (a) since kerhi is contained in kerhi. Assertions (b) and (c)* * follow from Corollary 6.7 (b). * * _|_| If one works with coefficients in Z(p), where p is a prime i_2+1 , the composi* *tion oeihi is an isomorphism according to Corollary 6.9 and one gets immediately: Corollary 6.11. For any ring R and any integer i 2 , the unstable Hurewic* *z homomorphism hi: Ki(R; Z(p)) ! Hi(E(R); Z(p)) is a split injection for all prime numbers p * *i_2+ 1 . Our second approach of the understanding of the Hurewicz homomorphism is based * *on the study of the relationships between its kernel and products in algebraic K-theory of the form ? : Ki(R) Kj(Z) -! Ki+j(R Z) ~=Ki+j(R) which have been defined in Definition 4.2 and Corollary 4.20. Theorem 6.12. For any ring R and any integer i 2 , the image of the product ho* *momorphism ? : Ki(R) K1(Z) -! Ki+1(R) is contained in the kernel of the unstable Hurewicz homomorphism hi+1: Ki+1(R) -! Hi+1(GL(R); Z) : Proof. Let us denote by KZ(-1) the (-1)-connected K-theory spectrum of Z , the * *0-th space of which is BGL(Z)+ x K0(Z) : it is a ring spectrum with unit j : S ! KZ(-1) whose 0-connec* *ted cover S(0) ! XZ is the map of spectra induced by the map of infinite loop spaces (B1 )+ ! BGL(Z* *)+ which comes from the obvious inclusion of the infinite symmetric group 1 into GL(Z) . This map j* * induces an isomorphism ~= j* : ss1(S) -! ss1(KZ(-1)) ~=K1(Z) and the image of j* : ssj(S) ! Kj(Z) for j * *2 is described in [67] and [75]. Let R be any ring and for i 2 , let us write XR(i - 1) for the (i - * *1)-connected cover of the 0-connected K-theory spectrum XR . It is obvious that ssj(XR(i - 1)) ~=Kj(R) fo* *r j i . By Definition 6.2 and Proposition 6.4, there is an exact sequence hi+2 i+2 ^ hi+1 Ki+2(R) -! Hi+2(XR(i - 1); Z) -! Ki(R) ss1(S) -! Ki+1(R) -! Hi+1(XR(i - 1)* *; Z) -! 0 : The diagram Ki(R) ss1(S)-^-! Ki+1(R) ? ? idj*?y~= ?y= Ki(R) K1(Z) -?-! Ki+1(R) ; which commutes since KR(-1) is a KZ(-1)-module, shows that the above exact sequ* *ence is actually hi+2 i+2 ? hi+1 Ki+2(R) -! Hi+2(XR(i - 1); Z) -! Ki(R) K1(Z) -! Ki+1(R) -! Hi+1(XR(i - 1);* * Z) -! 0 : Now, let us write BGL(R)+(i - 1) for the (i - 1)-connected cover of the infinit* *e loop space BGL(R)+ and 29 consider the commutative diagram Ki+1(R) hi+1--!Hi+1(BGL(R)+(i - 1); Z) ?? ? y = ?yoei+1 hi+1 Ki+1(R) --! Hi+1(XR(i - 1); Z) ; where the iterated homology suspension oei+1is an isomorphism since i 2 (see [* *100], Corollary VII.6.5). Thus, the composition Ki(R) K1(Z) -?!Ki+1(R) hi+1-!Hi+1(BGL(R)+(i - 1); Z) is trivial and the assertion immediately follows if one composes hi+1with the h* *omomorphism Hi+1(BGL(R)+(i - 1); Z) ! Hi+1(BGL(R)+; Z) ~=Hi+1(GL(R); Z) induced by the obvious map BGL(R)+(i - 1) ! BGL(R)+ . * * _|_| By an analogous argument, it is possible to generalize this result as follows. Theorem 6.13. If R is any ring, and if i and j are two integers such that i - 1* * j 1 , then the composition Ki(R) Kj(Z) -?!Ki+j(R) hi+j-!Hi+j(GL(R); Z) is trivial on all elements of the form x y with x 2 Ki(R) and y belonging to t* *he image of j* : ssj(S) ! Kj(Z) . Proof. See [14], Proposition 3.1. * * _|_| Remark 6.14. The assertions of Theorems 6.12 and 6.13 still hold if one replace* *s the infinite general linear group GL(R) by the group of elementary matrices E(R) or, if one assumes that i * * 3 , by the infinite Steinberg group St(R) (see [14], Proposition 3.1 and Theorem 3.2). In low dimensions, we are able to be more precise by providing exactness result* *s. For instance, let us describe the unstable Hurewicz homomorphism in dimension 3 . Theorem 6.15. For any ring R there is a natural exact sequence K2(R) K1(Z) -?!K3(R) h3-!H3(E(R); Z) -! 0 : Proof. (See [14], Theorem 4.1.) Let us consider the 1-connected infinite loop s* *pace BE(R)+ and kill all its homotopy groups above dimension 3 . We get its third Postnikov section (see als* *o Section 7) BE(R)+[3] which has only two non-trivial homotopy groups ss2(BE(R)+[3]) ~=K2(R) and ss3(B* *E(R)+[3]) ~=K3(R) . Therefore, BE(R)+[3] fits into the fibration of spaces K(K3(R); 3) -! BE(R)+[3] -! K(K2(R); 2) ; in which the base space and the fiber are Eilenberg-MacLane spaces. Similarly, * *look at the third Postnikov section XR(1)[3] of the 1-connected cover XR(1) of XR and at the cofibration of* * spectra 3H(K3(R)) -! XR(1)[3] -! 2H(K2(R)) ; 30 in which the base and the fiber are Eilenberg-MacLane spectra. This induces the* * following commutative diagram where both rows are homology exact sequences: . .-.-!H4(K(K2(R); 2); Z)@--!K3(R)-h3-! H3(BE(R)+; Z)--! 0 ?? ? ? y oe4 ?y~= ?yoe3 @ h3 . .-.-!H4(2H(K2(R)); Z)--! K3(R) --! H3(XR(1); Z)--! 0 : Here @ and @ are connecting homomorphisms and the three vertical arrows are ite* *rated homology suspen- sions. Because of the long stable Whitehead exact sequence (see Definition 6.2 * *and Proposition 6.4), it turns out easily that H4(2H(K2(R)); Z) ~=ss3(2H(K2(R)) ^ S(0)) ~=K2(R) ss1S ~=K2(R) K1(Z) and it is again possible to check that @ is the product ? : K2(R) K1(Z) ! K3(R* *) (see Proposition 2.2 of [14]). Since oe4 is surjective (see [100], Corollary VII.6.5) one can deduce* * that the image of @ is actually equal to the image of @, i.e., to the product K2(R) ? K1(Z) . * * _|_| A similar argument provides the next theorem on the unstable Hurewicz homomorph* *ism relating the algebraic K-theory of ring R to the homology of its infinite Steinberg group in dimension* *s 4 and 5 . Theorem 6.16. For any ring R there is a natural exact sequence K5(R) h5-!H5(St(R); Z) -! K3(R) K1(Z) -?!K4(R) h4-!H4(St(R); Z)!0 and the kernel of h5 fits into the natural exact sequence 0 -! K4(R) K1(Z) -?!kerh5- ! Q(R) -! 0 ; where Q(R) is a quotient of the subgroup of elements of order dividing 2 in K3(* *R) . Proof. See [14], Theorem 4.3. * * _|_| The last point of view from which we want to study the Hurewicz homomorphism is* * based on the Postnikov decomposition of CW-complexes. This is the suject of the next section. 7. The Postnikov invariants in algebraic K-theory The Postnikov invariants of a connected simple CW-complex X are cohomology clas* *ses which provide the necessary information for the reconstruction of X , up to a weak homotopy equiv* *alence, from its homotopy groups. Let ffi: X ! X[i] denote the i-th Postnikov section of X for any positi* *ve integer i : X[i] is the CW-complex obtained from X by killing the homotopy groups of X in dimensions > * *i , more precisely by adjoining cells of dimensions i + 2 such that ssj(X[i]) = 0 for j > i and (ffi* *)* : ssj(X) ! ssj(X[i]) is an isomorphism for j i . Thus, we may view X[i] as the i-th homotopical approx* *imation of X . The Postnikov k-invariants of X are cohomology classes ki+1(X) 2 Hi+1(X[i - 1]; ssi(X)) ; for i 2 , which are defined as follows (see for instance [100], Section IX.2). 31 Definition 7.1. Let X be a simple CW-complex, i an integer 2 , and let i+1deno* *te the composition "hi+1)-1 @ Hi+1(X[i - 1]; X[i]; Z) (-! ssi+1(X[i - 1]; X[i]) -! ssi(X[i]) ~* *=ssi(X) ; where "hi+1is the Hurewicz isomorphism for the i-connected pair (X[i - 1]; X[i]* *) and @ the connecting homomorphism (which is actually an isomorphism) of the homotopy exact sequence * *of that pair. Consider the isomorphism ~= i+1 : Hom(Hi+1(X[i - 1]; X[i]; Z); ssi(X)) -! H (X[i - 1]; X[i]; s* *si(X)) given by the universal coefficient theorem and the homomorphism : Hi+1(X[i - 1]; X[i]; ssi(X)) ! Hi+1(X[i - 1]; ssi(X)) induced by the inclusion of pairs (X[i - 1]; *) ,! (X[i - 1]; X[i]) . The k-inv* *ariant ki+1(X) is defined by ki+1(X) = (i+1) 2 Hi+1(X[i - 1|; ssi(X)) : The main property of these invariants is that X[i] is the homotopy fiber of the* * map X[i-1] ! K(ssi(X); i+1) corresponding to the cohomology class ki+1(X) 2 Hi+1(X[i - 1|; ssi(X)) , for i * * 2 . In other words, there is a commutative diagram of fibrations K(ssi(X); i)--'----! K(ssi(X); i) ?? ? y ?y X[i] - -----! PK(ssi(X); i + 1) ?? ? y ?yp i+1(X) X[i - 1] - k-----! K(ssi(X); i + 1) ; in which the right column is the path fibration over the Eilenberg-MacLane spac* *e K(ssi(X); i + 1) and the bottom square is a homotopy pull-back. Consequently, the knowledge of X[i - 1] * *, ssi(X) and ki+1(X) enables us to construct the next homotopical approximation X[i] of X . Remark 7.2. From that point of view, the understanding of the (weak) homotopy t* *ype of the K-theory space BGL(R)+ of a ring R depends on the knowledge of the K-groups Ki(R) = ssi(* *BGL(R)+) and of the k-invariants ki+1(BGL(R)+) . In the remainder of this section and in Section 9,* * we shall give some results on the k-invariants of K-theory spaces, especially on the (additive) order of the * *k-invariants ki+1(BGL(R)+) considered as elements of the group Hi+1(BGL(R)+[i - 1]; Ki(R)) . In order to understand the role of the k-invariants of a simple CW-complex, let* * us first mention the following obvious fact. Lemma 7.3. If ki+1(X) = 0 in Hi+1(X[i - 1]; ssi(X)) , then X[i] ' X[i - 1] x K(ssi(X); i) and the Hurewicz homomorphism hi: ssi(X) ! Hi(X; Z) is split injective. Proof. Since the diagram occuring in Definition 7.1 is a pull-back, the vanishi* *ng of ki+1(X) implies that X[i] = {(x; y) 2 X[i - 1] x PK(ssi(X); i + 1) | p(y) = *} ' X[i - 1] x (fiber * *ofp) ' X[i - 1] x K(ssi(X); i) : 32 By definition of ffi : X ! X[i] , the induced homomorphism (ffi)* : Hj(X; Z) ! * *Hj(X[i]; Z) is an iso- morphism for j i by the Whitehead theorem (see [100], Theorem IV.7.13). Thus, * *the K"unneth formula gives Hi(X; Z) ~=Hi(X[i]; Z) ~=Hi(X[i - 1]; Z) ssi(X) : * *_|_| One of the crucial properties of the k-invariants is the following lemma which * *follows almost directly from Definition 7.1 (see [100], Section IX.5, Example 3). Lemma 7.4. If X is a loop space X ' Y , then the k-invariants of X and Y are* * related by the formula oe*(ki+2(Y )) = ki+1(X) , where oe* : Hi+2(Y [i]; ssi(X)) ! Hi+1(X[i - * *1]; ssi(X)) is the cohomology suspension. Our first result is a vanishing theorem (see Theorem 7.6 below) based on the fo* *llowing remark on the cohomology suspension for Eilenberg-MacLane spaces. Proposition 7.5. For any abelian groups G and M , the double cohomology suspens* *ion (oe*)2: H5(K(G; 3); M) -! H4(K(G; 2); M) -! H3(K(G; 1); M) is trivial. Proof. For any abelian group G , it is known that H4(K(G; 3); Z) = 0 and it fol* *lows easily from Remark 6.3 that H5(K(G; 3); Z) ~=4(K(G; 3)) ~=G ss1(S) = G Z=2 ~=G=2G (see also [100* *], Theorems V.7.8 and XII.3.20). Thus, the universal coefficient theorem provides an isomorphism H5(K(G; 3); M) ~=Hom(G=2G; M) : For any element u 2 H5(K(G; 3); M) let us write bufor the corresponding element* * in Hom (G=2G; M) . For example, if one takes any abelian group G and M = G=2G , the element Sdq2co* *rresponding to the Steenrod square Sq2 viewed as a cohomology operation belonging to H5(K(G; 3* *); G=2G) turns out to be the identity id 2 Hom (G=2G; G=2G) . Now, for any cohomology class u 2 H* *5(K(G; 3); M) , it is clear that bu= bu](id) = bu](dSq2) , where bu]: Hom (G=2G; G=2G) ! Hom (G=2G* *; M) is induced by bu2 Hom(G=2G; M) . Consequently, u = bu*(Sq2) , where bu*: H5(K(G; 3); G=2G) ! * *H5(K(G; 3); M) is the homomorphism induced by bu. Finally, let us consider the commutative diagram H5(K(G; 3); G=2G)--bu*----!H5(K(G; 3); M) ?? ? y(oe*)2 ?y(oe*)2 H3(K(G; 1); G=2G)--bu*----!H3(K(G; 1); M) : Because it is well known that the cohomology operation (oe*)2(Sq2) is trivial i* *n H3(K(G; 1); G=2G) , we may deduce that (oe*)2(u) = (oe*)2(bu*(Sq2)) = bu*(oe*)2(Sq2) = 0 : * *_|_| Theorem 7.6. The first k-invariant k3(X) 2 H3(K(ss1(X); 1); ss2(X)) of any conn* *ected double loop space X is trivial. Proof. Consider any connected double loop space X ' 2Y . We may assume that Y * *is 2-connected and consequently that Y [3] ' K(ss3(Y ); 3) ' K(ss1(X); 3) . According to Lemma 7.4* *, k3(X) = (oe*)2(k5(Y )) , 33 where (oe*)2 is the double cohomology suspension (oe*)2: H5(Y [3]; ss2(X)) ~=H5(K(ss1(X); 3); ss2(X)) -! H3(X[1]; ss2(X)) ~=H* *3(K(ss1(X); 1); ss2(X)) : Therefore, the assertion is a direct consequence of Proposition 7.5. See [8] fo* *r another proof. _|_| Corollary 7.7. For any connected double loop space X , H2(X; Z) ~=ss2(X) 2(ss1(X)) where 2 denotes the exterior square. Proof. Since k3(X) = 0 in H3(K(ss1(X); 1); ss2(X)) by the previous theorem, the* * second Postnikov section X[2] of X is a product of Eilenberg-MacLane spaces according to Lemma 7.3: X[2] ' K(ss1(X); 1) x K(ss2(X); 2) : Thus, H2(X; Z) ~=H2(K(ss1(X); 1); Z) H2(K(ss2(X); 2); Z) : The second summand is isomorphic to ss2(X) by the Hurewicz theorem and the fact* * that X is an H-space implies that ss1(X) is abelian and consequently that H2(K(ss1(X); 1); Z) ~=2(ss* *1(X)) (see [37], Theorem V.6.4). * * _|_| A direct application of that result to the infinite loop space BGL(R)+ (see Rem* *ark 4.12) provides the following splitting (see also [11], Section 3, for the discussion of the natura* *lity of that splitting). Theorem 7.8. For any ring R , H2(GL(R); Z) ~=K2(R) 2(K1(R)) : Remark 7.9. This statement is quite obvious when the ring R is commutative with* * SK1(R) = 0 . In that case, E(R) = SL(R) , BSL(R)+ ' BGLg(R)+, K1(R) = Rx (see Theorem 2.9 and L* *emma 3.3), and there is a fibration of infinite loop spaces BSL(R)+ ! BGL(R)+ ! K(Rx; 1) w* *hich has a splitting induced by the inclusion Rx = GL1(R) ,! GL(R) . Therefore, BGL(R)+ ' BSL(R)+ xK* *(Rx; 1) and one gets the assertion. However, in the general case, the above topological argumen* *t involving k3(BGL(R)+) is necessary. This kind of nice consequences can be generalized when the k-invariant ki+1(X) * *is a cohomology class which is not trivial, but of finite order in the group Hi+1(X[i - 1]; ssi(X)) . Proposition 7.10. Let X be a connected simple CW-complex, i an integer 2 and a* *e a positive integer. The following assertions are equivalent: (a) ae ki+1(X) = 0 in Hi+1(X[i - 1]; ssi(X)) . (b) There is a map fi: X -! K(ssi(X); i) such that the induced homomorphism (fi)* : ssi(X) ! ssi(X) is multiplicatio* *n by ae . (c) There is a homomorphism i: Hi(X; Z) ! ssi(X) such that the composition ssi(X) hi-!Hi(X; Z) -i!ssi(X) is multiplication by ae . 34 Proof. (See also Section 1 of [16].) If (a) holds, the composition i+1(X) ae(id) X[i - 1] k-! K(ssi(X); i + 1) -! K(ssi(X); i + 1) (where id is written for the identity K(ssi(X); i + 1) ! K(ssi(X); i + 1) ) is * *trivial since it corresponds to the cohomology class ae ki+1(X) = 0 . Therefore, we have the following commutat* *ive diagram K(ssi(X); i)--'----! K(ssi(X); i)--ae(id)----!K(ssi(X); i) ?? ? ? y ?y ?y X[i] ------! PK(ssi(X); i +-1)ae(id)-----!PK(ssi(X); i + 1) ?? ? ? y ?yp ?yp i+1(X) ae(id) X[i - 1] -k-----! K(ssi(X); i + 1)------! K(ssi(X); i + 1) ; where all columns are fibrations and in which the bottom left square is a pull-* *back by definition of the k- invariant ki+1(X) . Let E be the pull-back of (ae ki+1(X) ; p) . Since the bott* *om composition in the above diagram is nullhomotopic, E is a product E ' X[i - 1] x K(ssi(X); i) . Since E * *is a pull-back, there is a map ' : X[i] ! E inducing an isomorphism on ssj for j i - 1 and multiplication* * by ae on ssi. Thus, we can define fi: X -ffi!X[i] -'!E ' X[i - 1] x K(ssi(X); i) -! K(ssi(X); i) ; where the last map is the projection onto the second factor. This map induces m* *ultiplication by ae on the only interesting homotopy group ssi: (fi)* : ssi(X) -.ae!ssi(X) : Assertion (c) follows from (b) because of the commutativity of the diagram ssi(X) --(fi)*----!.aessi(X) ?? ? y hi ?y~= Hi(X; Z) --(fi)*----!Hi(K(ssi(X);~i);=Z)ssi(X) induced by the map fi, where both vertical arrows are Hurewicz homomorphisms: w* *e call i the bottom horizontal homomorphism (fi)* in that diagram. In order to prove that (a) follows from (c), let us look at the commutative dia* *gram "hi+1 ssi+1(X[i - 1]; X[i])------!~=Hi+1(X[i - 1]; X[i]; Z) ? ? ~=?y@ ?ye@ ssi(X) ~=ssi(X[i]) ---hi---! Hi(X[i]; Z)~=Hi(X; Z) ; in which the horizontal arrows are Hurewicz homomorphisms and the vertical arro* *ws are connecting homo- morphisms. If i: Hi(X; Z) ! ssi(X) exists as in (c), we deduce that ie@= ihi@ ("hi+1)-1 = ae@ ("hi+1)-1 = aei+1; 35 where i+1is the element introduced in Definition 7.1. Thus, the image of aei+1u* *nder the isomorphism ~= i+1 : Hom(Hi+1(X[i - 1]; X[i]; Z); ssi(X)) -! H (X[i - 1]; X[i]; s* *si(X)) belongs to the image of the connecting homomorphism ffi : Hi(X[i]; ssi(X)) ! Hi* *+1(X[i - 1]; X[i]; ssi(X)) . The exactness of the cohomology sequence Hi(X[i]; ssi(X)) -ffi!Hi+1(X[i - 1]; X[i]; ssi(X)) -! Hi+1(X[i - 1]* *; ssi(X)) of the pair (X[i - 1]; X[i]) finally implies that ae ki+1= ae(i+1) = (aei+1) = * *0 . _|_| Because of these equivalences, it is really important to prove finiteness resul* *ts for the order of the k-invariants in algebraic K-theory. For that purpose, we first need to recall that H. Cartan* * computed the homology of Eilenberg-MacLane spaces in [38]; in particular, according to his calculatio* *n (see [38], Theoreme 2), the stable homotopy groups of Eilenberg-MacLane spaces have a quite small exponent.* * This can be formulated as follows. Definition 7.11. Let L1 := 1 , and for k 2 let Lk denote the product of all pr* *imes p for which there exists a sequence of non-negative integers (a1; a2; a3; : :):satisfying: (a) a1 0 mod (2p - 2) , ai 0 or 1 mod (2p - 2) for i 2 , (b) aiPpai+1for i 1 , (c) 1i=1ai= k . For example, L2= 2 , L3= 2 , L4= 6 , L5= 6 , L6= 2 , L7= 2 , L8= 30; : :.:Obser* *ve that Lk divides the product of all primes p k_2+ 1 . Lemma 7.12. For any abelian group G and any pair of integers i and m with 2 m * *< i < 2m , one has Li-mHi(K(G; m); Z) = 0 . Proof. This follows directly from Cartan's determination of the stable homology* * of Eilenberg-MacLane spaces given by Theoreme 2 of [38]. * * _|_| This implies the following consequence. Corollary 7.13. Let X be a (b-1)-connected CW-complex (with b 2 ) such that th* *ere exists an integer t b with the property that ssi(X) = 0 for i > t (in other words, such that X =* * X[t] ). Then Li-bLi-b-1Li-b-2. .L.i-tHi(X; Z) = 0 if t < i < 2b . Proof. Let i be an integer such that t < i < 2b . If t = b , then X is an Eil* *enberg-MacLane space X = K(ssb(X); b) and the result is given by Lemma 7.12. Now, let us suppose t >* * b . For any integer k with 1 k t - b , let us consider the fibration K(ssb+k(X); b + k) -! X[b + k] -! X[b + k - 1] ; whose Serre spectral sequence provides the exact sequence Hi(K(ssb+k(X); b + k); Z) -! Hi(X[b + k]; Z) -! Hi(X[b + k - 1]; * *Z) ; since i < 2b . Observe that for k = 1 , X[b + k - 1] = X[b] = K(ssb(X); b) and* * consequently that Li-bHi(X[b]; Z) = 0 according to Lemma 7.12. For the same reason, Li-b-kHi(K(ss* *b+k(X); b + k); Z) = 0 for 1 k t - b . We then conclude by induction that Li-bLi-b-1Li-b-2. .L.i-tHi* *(X[t]; Z) = 0 and get the assertion because X[t] = X by hypothesis. * * _|_| 36 Qj Definition 7.14. Let Rj:= 1 for j 1 and Rj:= k=2Lk for j 2 . For example, R2* * = 2 , R3 = 4 , R4 = 24 , R5 = 144 , R6 = 288 R7 = 576 , R8 = 17280; : :.:It turns out that a * *prime number p divides Rj if and only if p j_2+ 1 . This definition enables us to describe universal bounds for the order of the k * *-invariants of iterated loop spaces (see [6] and Section 1 of [10]). Let us emphasize the fact that the nex* *t result holds without any finiteness condition on the space we are looking at. Theorem 7.15. If X is a (b - 1)-connected r-fold loop space (with b 1 , r 0 )* *, i.e., X ' rY for some (b + r - 1)-connected CW-complex Y , then Ri-b+1ki+1(X) = 0 in Hi+1(X[i - 1]; ssi(X)) for all integers i such that 2 i r + 2b - 2 . Proof. Since X is (b-1)-connected, it is clear that ki+1(X) = 0 for 2 i b . T* *hus, we may assume that b + 1 i r + 2b - 2 , in particular that r + b 3 . It follows from the homoto* *py equivalence X ' rY that ssi(X) ~=ssi+r(Y ) and from Lemma 7.4 that the iterated cohomology suspens* *ion (oe*)r: Hi+r+1(Y [i + r - 1]; ssi(X)) -! Hi+1(X[i - 1]; ssi(X)) satisfies (oe*)r(ki+r+1(Y )) = ki+1(X) : Since we may assume that Y is (b + r - 1)-connected, we deduce from Corollary * *7.13 that Lj-b-rLj-b-r-1Lj-b-r-2. .L.j-i-r+1Hj(Y [i + r - 1]; Z) = 0 for i + r - 1 < j < 2b + 2r , in particular for j = i + r and j = i + r + 1 : Li-bLi-b-1Li-b-2._.L.1____-z________"Hi+r(Y [i + r - 1]; Z* *) = 0 ; =Ri-b Li-b+1Li-bLi-b-1._.L.2____-z________"Hi+r+1(Y [i + r - 1]; * *Z) = 0 : =Ri-b+1 Therefore, the universal coefficient theorem shows that the exponent of the gro* *up Hi+r+1(Y [i+r-1]; ssi(X)) is bounded by lcm(Ri-b; Ri-b+1) = Ri-b+1. Thus, Ri-b+1ki+r+1(Y ) = 0 and Ri-b+1ki+1(X) = Ri-b+1(oe*)r(ki+r+1(Y )) = (oe*)r(Ri-b+1ki+r+1(Y ))* *_=|0|:_ Corollary 7.16. For any (b - 1)-connected infinite loop space X (with b 1 ), Ri-b+1ki+1(X) = 0 in Hi+1(X[i - 1]; ssi(X)) for all integers i 2 . In the case of the K-theory spaces, we get the following result. Theorem 7.17. For any ring R , (a) Riki+1(BGL(R)+) = 0 for all i 2 , (b) Ri-1ki+1(BE(R)+) = 0 for all i 3 , (c) Ri-2ki+1(BSt(R)+) = 0 for all i 4 . Proof. This follows from Corollary 7.16, because BGL(R)+ , BE(R)+ and BSt(R)+ * *are infinite loop spaces which are connected, simply connected and 2-connected respectively. * * _|_| 37 Let us look at immediate consequences of this theorem for the Hurewicz homomorp* *hism relating the K- groups of any ring R to the homology of the infinite general linear group over * *R , respectively of the infinite special linear group and of the infinite Steinberg group. Remember tha* *t this homomorphism is an isomorphism in the first non-trivial dimension (see Theorem 6.1). Our next resu* *lt approximates the exponent of the kernel of the Hurewicz homomorphism in all dimensions (see also [9]). Remark 7.18. In [87], Proposition 3, C. Soule has shown that the kernel of hi: * *Ki(R) ! Hi(E(R); Z) is a torsion group that involves only prime numbers p satisfying p i+1_2, but his* * argument does not imply that this kernel has finite exponent. Corollary 7.19. Let R be any ring. (a) For any i 2 , the Hurewicz homomorphism hi: Ki(R) ! Hi(GL(R); Z) satisfies* * Ri(kerhi) = 0 . (b) For any i 3 , the Hurewicz homomorphism hi: Ki(R) ! Hi(E(R); Z) satisfies * *Ri-1(kerhi) = 0 . (c) For any i 4 , the Hurewicz homomorphism hi: Ki(R) ! Hi(St(R); Z) satisfies* * Ri-2(kerhi) = 0 . Proof. Let us start with the 0-connected infinite loop space BGL(R)+ . Because * *of Proposition 7.10 and of Corollary 7.16, there is a homomorphism i: Hi(GL(R); Z) ~=Hi(BGL(R)+; Z) ! K* *i(R) such that the composition Ki(R) hi-!Hi(GL(R); Z) -i!Ki(R) is multiplication by Ri. If x belongs to the kernel of hi, then Rix = ihi(X) = * *0 . The same argument works for BE(R)+ and BSt(R)+ . * * _|_| Remark 7.20. Of course, the assertion (a) is less interesting than the other on* *es since it can be improved: for instance, in the case where i = 2 , we know from Theorem 7.8 that h2: K2(R)* * ! H2(GL(R); Z) is split injective for any ring R . Example 7.21. Corollary 7.19 shows that h3: K3(R) ! H3(E(R); Z) fulfills 2 (kerh3) = 0 for any ring R . This was first observed by A.A. Suslin in [91], Proof of Propo* *sition 4.5 (no details are given there). Later, C.H. Sah has also established that 2 kerh3= 0 for any ring A (se* *e [80], Proposition 2.5), but unfortunately, there is a gap in his proof (see [9], Remark 1.9). Corollary 7.16 also provides another proof of Corollary 6.11. Corollary 7.22. Let R be any ring. (a) For any integer i 1 , the Hurewicz homomorphism hi : Ki(R; Z(p)) ! Hi(GL(R* *); Z(p)) is a split injection for all primes p i+3_2. (b) For any integer i 2 , the Hurewicz homomorphism hi : Ki(R; Z(p)) ! Hi(E(R)* *; Z(p)) is a split injection for all primes p i+2_2. (c) For any integer i 3 , the Hurewicz homomorphism hi : Ki(R; Z(p)) ! Hi(St(R* *); Z(p)) is a split injection for all primes p i+1_2. Proof. Let us look again at the composition Ki(R) hi-!Hi(GL(R); Z) -i!Ki(R) which is multiplication by Ri. Since Ri is only divisible by primes p i+2_2, t* *he composition Ki(R; Z(p)) hi-!Hi(GL(R); Z(p)) -i!Ki(R; Z(p)) 38 is an isomorphism and hi is a split injection when p i+3_2. The proof is analo* *gous for the simply connected infinite loop space BE(R)+ (with p dividing Ri-1if and only if p i+1_2) and fo* *r the 2-connected infinite loop space BSt(R)+ (with p dividing Ri-2if and only if p i_2). * * _|_| Let us conclude this section by mentioning a result on the homotopy type of the* * K-theory space of alge- braically closed fields (see also [9], Theorem 2.4). Theorem 7.23. Let F be an algebraically closed field and i any positive even in* *teger. Then, (a) the Postnikov k-invariant ki+1(BSL(F)+) is trivial in Hi+1(BSL(F)+[i - 1]; * *Ki(F)) , (b) the Hurewicz homomorphism hi: Ki(F) ! Hi(SL(F); Z) is split injective. Proof. Since BSL(F)+ is a simply connected infinite loop space, we may consider* * an (i - 1)-connected space Y with BSL(F)+ ' i-2Y . By Lemma 7.4, the k-invariant ki+1(BSL(F)+) is t* *hen the image of k2i-1(Y ) under the (i - 2) -fold iterated cohomology suspension (oe*)i-2: H2i-1(Y [2i - 3]; Ki(F)) ! Hi+1(BSL(F)+[i - 1]; Ki(F))* * : Now, look at the universal coefficient theorem H2i-1(Y [2i - 3]; Ki(F)) ~=Hom(H2i-1(Y [2i - 3]; Z); Ki(F)) Ext(H2i-2(Y [* *2i - 3]; Z); Ki(F)) ; and observe that the group Ext(H2i-2(Y [2i - 3]; Z); Ki(F)) vanishes because A.* *A. Suslin proved in Section 2 of [92] that Ki(F) is divisible for algebraically closed fields. Moreover, he* * also obtained in [92], Section 2, that Ki(F) is torsion-free if i is an even integer: this and the fact that * *H2i-1(Y [2i - 3]; Z) is a torsion group (see Corollary 7.13) imply that Hom (H2i-1(Y [2i - 3]; Z); Ki(F))* * is trivial. Consequently, ki+1(BSL(F)+) = (oe*)i-2(k2i-1(Y )) vanishes because k2i-1(Y ) 2 H2i-1(Y [2i - * *3]; Ki(F)) = 0 . Assertion (b) follows from Lemma 7.3. * * _|_| 8. The algebraic K-theory of number fields and rings of integers In the remainder of the paper, let us concentrate our attention on a specific c* *lass of rings: we want to investigate the K-groups of number fields and rings of integers. This plays an * *important role because of the various interactions between algebraic K-theory and number theory. Let F be a n* *umber field (i.e., a finite extension of the field of rationals Q ) and OF its ring of algebraic integers. * *D. Quillen obtained in 1973 the first result on the structure of the groups Ki(OF) (see [73], Theorem 1). Theorem 8.1. (Quillen) For any number field F and for any integer i 0 , Ki(OF)* * is a finitely generated abelian group. The corresponding result does not hold for the number field F itself: the struc* *ture of the abelian groups Ki(F) is much more complicated, and consequently much more interesting. Of cour* *se, the groups Ki(F) and Ki(OF) are strongly related. In order to observe that relation, D. Quillen * *constructed in Sections 5 and 7 of [72] (see also [74], Theorem 4) a fibration Y BQP(OF=m ) -! BQP(OF) -! BQP(F) ; m Q where is the weak product (i.e., the direct limit of cartesian products with * *finitely many factors), where m runs over the set of all maximal ideals of OF and where the last map is induced* * by the inclusion OF ,! F . Here, for any ring R , P(R) is the category of finitely generated projective R-* *modules and BQP(-) denotes 39 the Q-construction mentioned in Remark 3.16: in particular, its loop space fulf* *ills the homotopy equivalence BQP(R) ' BGL(R)+ x K0(R) . By looping the base space and the total space of the* * above fibration and by taking the 0-connected covers of the the three spaces, we get the fibration Y BGL(OF)+ -! BGL(F)+ -! -1BGL(OF=m )+ : m The homotopy exact sequence of that fibration provides the following long exact* * sequence. Theorem 8.2. For any number field F , there is a long exact sequence (called th* *e localization sequence in algebraic K-theory) M . .-.! Ki(OF) -! Ki(F) -! Ki-1(OF=m ) -! Ki-1(OF) -! Ki-1(F) -! . .* * . m M . .-.! K1(OF) -! K1(F) -! K0(OF=m ) -! K0(OF) -! K0(F) ; m where m runs over the set of all maximal ideals of OF . Moreover, C. Soule could improve this result by showing that this long exact se* *quence breaks into short exact sequences for all positive integers i (see [86], Theoreme 1): M 0 -! Ki(OF) -! Ki(F) -! Ki-1(OF=m ) -! 0 : m Since OF=m is a finite field, the vanishing of Kj(OF=m ) whenever j is even 2 * *(see Corollary 5.9) then implies the following result. Theorem 8.3. Let F be any number field. (a) For any odd integer i 3 , the inclusion OF ,! F induces an isomorphism ~= Ki(OF) -! Ki(F) ; (b) For any even integer i 2 , there is a short exact sequence M 0 -! Ki(OF) -! Ki(F) -! Ki-1(OF=m ) -! 0 ; m where m runs over the set of all maximal ideals of OF and where Ki-1(OF=m )* * can be determined by Corollary 5.9. Remark 8.4. Similar results hold for rings of S-integers in F , where S is any * *set of places of F (see [86], Theoreme 1). The following finiteness result follows immediately from Theorems 8.1 and 8.3. Corollary 8.5. For any number field F and any odd integer i 3 , the group Ki(F* *) is finitely generated. The next important information on the structure of the K-groups of number field* *s and rings of integers was obtained by A. Borel as a consequence of his study of the real cohomology of li* *near groups (see [31] or [32], Section 11). 40 Theorem 8.6. (Borel) Let F be a number field and let us write [F : Q] = r1+ 2r* *2, where r1 is the number of distinct embeddings of F into R and r2 the number of distinct conjuga* *te pairs of embeddings of F into C with image not contained in R . (a) If R denotes either the number field F or its ring of algebraic integers OF* * , then the rational coho- mology of the special linear group SL(R) is given by O O H*(SL(R); Q) ~=( Aj) ( Bk) ; 1jr1 1kr2 where j runs over all distinct embeddings of F into R , k over all distinct* * conjugate pairs of embeddings of F into C with image not contained in R , and where Aj and Bk are the fol* *lowing exterior algebras: Aj= Q(x5; x9; x13; : :;:x4l+1; : :):and Bk = Q(x3; x5; x7; : :;:x* *2l+1; : :): with deg(xj) = j . (b) If R denotes either the number field F or its ring of algebraic integers OF* * , then for any integer i 2 , (0 , if i is even, Ki(R) Q ~= Qr1+r2 , if i 1 mod 4, Qr2 , if i 3 mod 4. As a consequence, we observe: Corollary 8.7. If R denotes a number field F or its ring of integers OF , then * *Ki(R) is a torsion group for all even integers i 2 . In order to summarize Theorem 8.1, Corollaries 8.5 and 8.7, we can formulate th* *e following statement. Corollary 8.8. Let F be any number field, OF its ring of integers and i a posit* *ive integer. (a) Ki(F) is finitely generated if i is odd and Ki(F) is a torsion group if i i* *s even. (b) Ki(F)=torsionis a free abelian group of finite rank (which is known by Theo* *rem 8.6 (b)) for all positive integers i . (c) Ki(OF) is finitely generated if i is odd and Ki(OF) is finite if i is even. However, the structure of the groups Ki(F) is quite complicated. In order to il* *lustrate this, let us consider the subgroup Di(F) of Ki(F) consisisting of all (infinitely) divisible elements* * in Ki(F) (notice that Di(F) is not necessarily a divisible subgroup of Ki(F) ) and prove the following surp* *rising assertion. Theorem 8.9. For any number field F , Di(F) = 0 if i is an odd integer 1 and D* *i(F) is a finite abelian group if i is an even integer 2 . Proof. Since Ki(F) is a finitely generated abelian group when i is odd accordin* *g to Corollary 8.5, it does not contain any non-trivial divisible element and Di(F) = 0 . When i is even, c* *onsider again the localization exact sequence M 0 -! Ki(OF) -! Ki(F) -! Ki-1(OF=m ) -! 0 : m By Corollary 5.9, Ki-1(OF=m ) is a finite cyclic groupLand contains therefore n* *o non-trivial divisible elements. Consequently, the same is true for the direct sum m Ki-1(OF=m ) . It then fo* *llows that all divisible elements in Ki(F) actually belong to the image of the homomorphism Ki(OF) ! Ki(* *F) induced by the inclusion OF ,! F . Finally, the fact that Ki(OF) is finite by Corollary 8.8 (c* *) shows that Di(F) is finite. (The elements of Di(F) can be viewed as elements of Ki(OF) , even if they are d* *ivisible only in Ki(F) .) _|_| 41 Remark 8.10. The situation is indeed quite strange because the group Di(F) does* * not vanish in general. This was proved in a very precise way by G. Banaszak in [23], Section VIII, and* * [24], Section II. For any odd prime p , let us write Di(F)p for the subgroup of p-torsion divisible elements * *in Ki(F) (in other words, Di(F)p is the p-component of Di(F) ). If F is a totally real number field, i = * *2m an even integer with m odd, G. Banaszak, together with M. Kolster, determined the order of the subgr* *oup D2m(F)p (see [24], Theorem 3): the order of D2m(F)p is exactly the p-adic absolute value of wm+1(F)_iF(-m)_Q; v|pwm (Fv) where iF(-) is the Dedekind zeta function of F , wm (k) the biggest integer s s* *uch that the exponent of the Galois group Gal(k(s)=k) divides m for any field k (here s is an s-th primi* *tive root of unity), and Fv the completion of F at v . For the case where F is the field of rationals Q * *, look at Remark 9.15 for a more explicit description of the order of the groups D2m(Q)p. Notice that the knowledge of D2m(F) is of particular interest since it is relat* *ed to the Lichtenbaum-Quillen conjecture in algebraic K-theory (see Remark 9.16 and [24], Section II.2) and t* *oetale K-theory (see [25], Section 3). In order to have an almost complete picture of the complexity of the structure * *of the algebraic K-groups of number fields, let us try to get analogous results for integral homology. Of* * course, the Hurewicz theorem modulo the Serre class of finitely generated abelian groups (see [82], Sections* * III.1 and III.2) for the space BSL(OF)+ enables us to deduce from Theorem 8.1 the following structure theorem * *for the integral homology of the special linear group over a ring of integers. Corollary 8.11. For any number field F and any integer i 0 , Hi(SL(OF); Z) is * *a finitely generated abelian group. The situation is more complicated for the special linear group SL(F) over the n* *umber field F itself: in fact, the structure of the groups Hi(SL(F); Z) turns out to be similar to the s* *tructure of Ki(F) described in Corollary 8.8 (b). Theorem 8.12. For any number field F and any integer i 0 , the group Hi(SL(F);* * Z) is the direct sum of a torsion group and a free abelian group of finite rank (which can be calcul* *ated by Theorem 8.6 (a)). Proof. (See also [5], Section 2.) The proof is based on the results on the Post* *nikov invariants described in Section 7. Let C denote the Serre class of all abelian torsion groups. Bec* *ause BSL(F)+ is a sim- ply connected infinite loop space, Corollary 7.16 implies that all Postnikov k-* *invariants of BSL(F)+ are cohomology classes of finite order. Therefore, Proposition 7.10 (b) provides a * *map 1Y f : BSL(F)+ -! K(Kj(F); j) j=2 which induces multiplication by the (finite) order of the corresponding k-invar* *iant ki+1(BSL(F)+) on each homotopy group ssi(BSL(F)+) ~=Ki(F) , i 2 . In particular, f induces a C-* *isomorphism on each homotopy group. Now, let us compose f with the natural map 1Y 1Y K(Kj(F); j) -! K(Kj(F)=torsion; j) j=2 j=2 42 which induces the quotient map (and thus a C-isomorphism) on each homotopy grou* *p. If we denote by Y this later space, this composition is a map 1Y : BSL(F)+ -! Y = K(Kj(F)=torsion; j) j=2 inducing a C-isomorphism on all homotopy groups and therefore also a C-isomorph* *ism * : Hi(BSL(F)+; Z) ! Hi(Y ; Z) on all integral homology groups because of the mod C Whitehead theorem (see [82* *], Section III.4). On the other hand, since ssi(Y ) ~= Ki(F)=torsionis finitely generated for all in* *tegers i 1 by Corol- lary 8.8 (b), the homology groups Hi(Y ; Z) of Y are also finitely generated. * * Consequently, the group Hi(BSL(F)+; Z)=ker * ~=image * is also finitely generated and ker * belongs to * *C. If Ti is written for the torsion subgroup of Hi(BSL(F)+; Z) , it follows that Hi(BSL(F)+; Z)=Ti is f* *initely generated, i.e., free abelian of finite rank, since it is a quotient of Hi(BSL(F)+; Z)=ker *. Finally* *, this implies the vanishing of Ext(Hi(BSL(F)+; Z)=Ti; Ti) and the splitting of the extension 0 -! Ti-! Hi(BSL(F)+; Z) -! Hi(BSL(F)+; Z)=Ti-! 0 : The assertion then follows from the isomorphism Hi(SL(F); Z) ~=Hi(BSL(F)+; Z) .* * _|_| Because of Theorem 8.3 (b), the groups Ki(F) are in general not finitely genera* *ted. By Serre class theory (see [82], Chapitre I), this implies that the homology groups Hi(SL(F); Z) are in ge* *neral not finitely generated. However, this only happens because of their torsion subgroups. The next step wo* *uld be to investigate the structure_of the torsion subgroups of the groups Hi(SL(F); Z) . In particular, * *let us look at the subgroup D i(F) of divisible elements in Hi(SL(F); Z) . Here again, an argument similar * *to the proof of Theorem 8.12 shows that this subgroup is relatively small in the following sense. * *__ Theorem 8.13. For any number field F and any integer i 0 , the abelian group * *D i(F) is of finite exponent. Proof. See [17], Theorem 1.1. * * _|_| Remark 8.14. Observe that for any number field F , the homomorphism Ki(OF) ! K* *i(F) induced by the inclusion OF ,! F is always injective according to Theorem 8.3. The anal* *ogous assertion for the induced homomorphism Hi(SL(OF); Z) ! Hi(SL(F); Z) is not true (see Remark 2.7 o* *f [18]). However, one can prove (see Theorem 1.4 of [18]) the injectivity of the induced homomorp* *hism Hi(SL(OF); Z(p)) ! Hi(SL(F); Z(p)) in small dimensions, more precisely for 2 i min(2p - 2; dp(F)* * + 1) , where dp(F) denotes the smallest positive integer j for which Kj(F) contains non-trivial p-* *torsion divisible elements ( dp(F) is an even integer according to Theorem 8.9 and we say that dp(F) = 1 i* *f there are no p-torsion divisible elements in Kj(F) for all j 1 ). 9. The algebraic K-theory of the ring of integers Z If we apply the results of the previous sections to the special case of the rin* *g of integers Z , we first know that E(Z) = SL(Z) by Theorem 2.10 and we may deduce from Section 8 the following res* *ult on the structure of the abelian groups Ki(Z) . 43 Theorem 9.1. For any positive integer i , aeZ finite group, if i 1 mod 4, i 5, Ki(Z) = finite group , otherwise. Proof. Theorem 8.1 asserts that the groups Ki(Z) are finitely generated abelia* *n groups for all i 0 . Moreover, the rank of the free abelian group Ki(Z)=torsionis given by Theorem 8* *.6 (b), with r1 = 1 and r2= 0 . * * _|_| In low dimensions, the K-groups of Z have been computed for i 4 and partially * *determined for i = 5 . Theorem 9.2. K0(Z) ~=Z , K1(Z) ~=Z=2 , K2(Z) ~=Z=2 , K3(Z) ~=Z=48 , K4(Z) = 0 * *and K5(Z) ~=Z (3-torsion finite.group) Proof. See Theorem 1.2, Theorem 2.10 and Example 2.29 for the calculation of K0* *(Z) , K1(Z) and K2(Z) , [55] for the determination of K3(Z) , [76], [88], [77], [99] and [95] for the v* *anishing of K4(Z) , and [56] and [84] for the description of K5(Z) . * * _|_| Remark 9.3. The cyclic groups of order 2 in Ki(Z) for i = 1 and i = 2 occur act* *ually in all dimensions i 1 or2 mod 8 , as observed by D. Quillen in [75]. Let us also look at the unstable Hurewicz homomorphisms hi: Ki(Z) ~=ssi(BSL(Z)+) -! Hi(BSL(Z)+; Z) ~=Hi(SL(Z); Z) for i 2 and hi: Ki(Z) ~=ssi(BSt(Z)+) -! Hi(BSt(Z)+; Z) ~=Hi(St(Z); Z) for i 3* * : Theorem 9.4. The following sequences are exact: (a) . .-.! K4(Z) h4-!H4(SL(Z); Z) -! Z=4 -! K3(Z) h3-!H3(SL(Z); Z) -! 0 , where kerh3~=K2(Z) ? K1(Z) ~=Z=2 , (b) . .-.! K5(Z) h5-!H5(St(Z); Z) -! K3(Z)__K1(Z)_-z_____"?-!K4(Z) h4-!H4(St(Z)* *; Z) -! 0 : ~=Z=2 Proof. The unstable Whitehead exact sequence (see [102]) of the simply connecte* *d space BSL(Z)+ is . .-.! K4(Z) h4-!H4(BSL(Z)+; Z) -! 3(BSL(Z)+) -! K3(Z) h3-!H3(BSL(Z)+; Z) * *-! 0 and 3(BSL(Z)+) ~=(ss2(BSL(Z)+)) ~=(K2(Z)) ~=(Z=2) ~=Z=4 ; where (-) is the quadratic functor defined on abelian groups by J.H.C. Whitehea* *d in Section 5 of [102] (see also [4], Satz 1.5). This gives the exact sequence (a) and Theorem 6.15 sh* *ows that the kernel of h3 is exactly the image of the product K2(Z) K1(Z) -?!K3(Z) . Assertion (b) follows * *directly from Theorem 6.16. * * _|_| Of course, this also produces (co)homological results. According to Remark 7.9 * *(or Lemma 1.2 of [4]), we have the homotopy equivalence BGL(Z)+ ' BSL(Z)+ x BZ=2 and it is therefore sufficient to investigate the homology of the universal cov* *er BSL(Z)+ of BGL(Z)+ . Let us first recall Theorem 8.6 (a) on the rational cohomology of SL(Z) . 44 Theorem 9.5. H*(SL(Z); Q) ~=Q(x5; x9; x13; : :x:4l+1; : :):, where deg(x4l+1) * *= 4l + 1 . We may determine the homology of SL(Z) and St(Z) in small dimensions from Theor* *ems 9.2 and 9.3 (see also [7] for the relations between H*(SL(Z); Z) and H*(St(Z); Z) ). Theorem 9.6. H2(SL(Z); Z) ~=Z=2 , H3(SL(Z); Z) ~=Z=24 , H4(SL(Z); Z) ~=Z=2 , H3(St(Z); Z) ~=Z=48 , H4(St(Z); Z) = 0 , H5(St(Z); Z) ~=Z (3-torsion finite gr* *oup)and there is a short exact sequence 0 -! K5(Z) h5-!H5(St(Z); Z) -! Z=2 -! 0 ; in which h5 is an isomorphism on the torsion subgroup of K5(Z) and multiplicati* *on by 2 on the infinite cyclic summand of K5(Z) . Proof. Theorems 6.1 and 9.2 imply that H2(SL(Z); Z) ~=K2(Z) ~=Z=2 and that H3(S* *t(Z); Z) ~=K3(Z) ~= Z=48 . It follows from the vanishing of K4(Z) and Theorem 9.4 that H3(SL(Z); Z* *) ~=Z=24 (see also [4], Satz 1.5), H4(SL(Z); Z) ~=Z=2 and H4(St(Z); Z) = 0 . Finally, it is possib* *le to show that the term Q(Z) occuring in Theorem 6.16 is trivial (see [13], Theorem 3, or [14], Proposi* *tion 5.1). Consequently, h5: K5(Z) ! H5(St(Z); Z) is injective because K4(Z) = 0 and we get the desired * *exact sequence. Finally, the effect of h5 on the infinite cyclic summand of K5(Z) is explained by Theore* *m 1.5 of [7]. _|_| The first more general result on the torsion of the the algebraic K-groups of Z* * has been obtained by D. Quillen in 1976 (see [75]). Remember that he had already computed the K-theory* * of finite fields. By studying the map Ki(Z) ! Ki(Fp) induced by the reduction mod p: Z ! Fp for var* *ious primes p , he could prove the following relationship between the order of the torsion subgrou* *ps of the K-groups of Z and the denominators of the Bernoulli numbers. Definition 9.7. The Bernoulli numbers are the rational numbers Bm occuring in t* *he power series __t__= 1 + 1X Bm_tm et- 1 m=1 m! of the complex function f(t) = __t__et-.1It is not hard to check that B1= -1_2and Bm = 0 for m odd 3 : The first Bernoulli numbers are B2= 1_6; B4= -_1_30; B6= 1_42; B8= -_1_30; B10= 5_66; B12= -_691_2730; B14= 7_6; B16= -3617_510; B18= 43867_798: i B j Definition 9.8. For any positive even integer m , let Em = denominator_m_m. For* * instance, E2= 12 ; E4= 120 ; E6= 252 ; E8= 240 ; E10= 132 ; E12= 32760 ; E14= 12 ; E16= 8160 ; E18= 14364 : It turns out that the numbers Em are completely determined by the following pro* *perty observed by K. von Staudt in 1845. 45 Lemma 9.9. Let p be a prime and m a positive even integer. Then, for s 1 , ps * *divides Em if and only if (p - 1)p(s-1)divides m . Proof. See [34], p.410, Satz 4, [98], p.56, Theorem 5.10, or [66], Appendix B, * *Theorem B.4. _|_| D. Quillen exhibited the following torsion classes in the algebraic K-theory of* * Z . Theorem 9.10. (Quillen) For any positive even integer j , the group K4j-1(Z) co* *ntains a cyclic subgroup Q4j-1of order 2E2j. If j is even, Q4j-1is a direct summand of K4j-1(Z) . If j i* *s odd, the odd-torsion part of Q4j-1is a direct summand of K4j-1(Z) and the 2-torsion part of Q4j-1(wh* *ich is ~=Z=8 ) is contained in a cyclic direct summand of order 16 of K4j-1(Z) . Proof. See [75] for the detection of the subgroup Q4j-1of order 2E2j in K4j-1(Z* *) and the discussion of the case where j is even, and Theorem 4.8 of [36] for the case j odd. * * _|_| We then may conclude the next consequence from Lemma 9.9 and Theorem 9.10. Corollary 9.11. The group Ki(Z) contains a cyclic subgroup of order 16 if i 3 * *mod 8 and of order 2(i + 1)2 if i 7 mod 8 , where (i + 1)2 denotes the 2-primary part of the inte* *ger (i + 1) . Another very surprising result was proved by C. Soule in 1979 when he explained* * that in fact the numerators of the Bernoulli numbers also play a role in the investigation of the torsion i* *n the groups Ki(Z) . Recall the following definition. Definition 9.12. A prime number p is called irregular if there exists a positiv* *e even integer m such that p divides the numerator of Bm_m(see [34], p.393-414, or [98], p.6 and Section 5* *.3, for more details). For instance, 691 is an irregular prime since B12_12= -_691_32760. It turns out tha* *t p is irregular if and only if p divides the class number of the cyclotomic field Q(p) , where p is a p-th primi* *tive root of unity; moreover, an irregular prime p is called properly irregular if p does not divide the clas* *s number of the maximal real subfield of Q(p) (see [98], p.39 and p.165). A regular prime is a prime number* * which is not irregular. Notice that there are infinitely many irregular primes but that it is still not* * known whether there are finitely or infinitely many regular primes. Theorem 9.13. (Soule) If p is a properly irregular prime number and if m is a p* *ositive even integer < p such that p divides the numerator of _Bm+1_m,+t1hen the algebraic K-theory * *group K2m(Z) contains an element whose order is equal to the p-primary part of Bm+1_m.+ 1 Proof. See [85], Section IV.3, Theoreme 6, where the argument is based on the i* *nvestigation of the relation- ships between algebraic K-theory andetale cohomology. * * _|_| Example 9.14. The group K22(Z) contains 691-torsion. Remark 9.15. Consider again the short exact sequences M 0 -! K2m(Z) -! K2m(Q) -! K2m-1(Z=p) -! 0 p prime given by Theorem 8.3 for all positive integers m . The torsion elements of K2m(* *Z) detected by C. Soule and presented in Theorem 9.13 also belong to the group K2m(Q) and play a specia* *l role in that group with 46 respect to the subgroup D2m(Q) of divisible elements in K2m(Q) (see Theorem 8.9* * and Remark 8.10). In fact G. Banaszak determined the precise order of the p-primary component D2m(Q)* *p of D2m(Q) as follows (see [24], Theorem 3): if m is an odd integer and p an odd prime, then the orde* *r of D2m(Q)p is equal to the p-adic absolute value of the numerator of Bm+1_m.+F1or example, D22(Q) is c* *yclic of order 691 . Remark 9.16. The torsion in the groups Ki(Z) is really mysterious and contains * *a lot of number theoretical information. It is the object of the following (still open) conjecture due to S* *. Lichtenbaum and D. Quillen: if m is a positive even integer, then the quotient of the order of K2m-2(Z) by the* * order of K2m-1(Z) should be equal to the absolute value of Bm_m, up to a power of 2 (see [57], [85], Sec* *tion I.1, or [58], p.102-103). Thus, apart from some classes of order 2, the known torsion classes in the alge* *braic K-theory of Z which occur in odd degrees, respectively in even degrees, are related to the denomina* *tors, respectively to the numerators, of the Bernoulli numbers. The next attempt to understand the K-theory of Z was made by M. B"okstedt in 19* *84 (see [30]): he tried to construct a model for the algebraic K-theory space BGL(Z)+ and proved that t* *his model detects the known torsion classes at the prime 2 . His idea was simple and excellent: he c* *onsidered the classifying space BO of the orthogonal group, the classifying space BU of the unitary group* * and, for a prime p , the K-theory space BGL(Fp)+ . Then, he introduced a space J(p) which is defined as * *the pull-back of the following diagram: __0 J(p) ------! BO ??_ ? yf0p ?y_c _ BGL(Fp)+ ---b---! BU ; _ where _cis the complexification and bthe composition of the plus construction o* *f the Brauer lift BGL(Fp) ! Fp with the inclusion ' : Fp ,! BU (see Definitions 5.1 and 5.7). Observe that * *the homotopy fiber of both horizontal maps in the above diagram is BU ' U . A direct calculation show* *s that ss1J(p) ~=ZZ=2 and we know that K1(Z) ~=Z=2 . Consequently, let us write JK(Z; p) for the cov* *ering space of J(p) corresponding to the factor Z=2 . It turns out that JK(Z; p) is the pull-back o* *f the following diagram: 0 JK(Z; p) ------! BO ?? ? y f0p ?yc BSL(Fp)+ ----b--! BSU ; where the bottom arrow is the universal cover of the corresponding line in the * *previous diagram. In order to approximate the 2-torsion of the K-groups of Z , M. B"okstedt chose a prime p * *3 or5 mod 8 , completed all spaces at 2, and constructed a map : (BGL(Z)+)^2! JK(Z; p)^2for which he * *was able to prove: Theorem 9.17. (B"okstedt) The map : (BGL(Z)+)^2! JK(Z; p)^2is a retraction. I* *n particular, the map : (BGL(Z)+)^2! JK(Z; p)^2induces a split surjection on all homotopy g* *roups. Proof. See [30], Theorem 2. * * _|_| 47 Observe at that point that it is easy to compute the homotopy groups of the spa* *ce JK(Z; p) : consequently, B"okstedt's theorem provides actually a direct summand_of each group Ki(Z) Z^2* *, where Z^2denotes the ring of 2-adic integers. Notice that induces a map : (BGL(Z)+xS1)^2! J(p)^2* *and that the localization exact sequence (see Theorem 8.3 and Remark 8.4) gives the following short exact* * sequence for all integers i 1 : 0 -! Ki(Z) -! Ki(Z[1_2]) -! Ki-1(F2) -! 0 : However, Ki-1(F2)Z^2is always trivial according to_Corollary 5.9 except if i = * *1 , where K0(F2)Z^2~=Z^2. Therefore, (BGL(Z)+ x S1)^2~=(BGL(Z[1_2])+)^2and is actually a map (BGL(Z[1_2* *])+)^2! J(p)^2which induces also a split surjection on all homotopy groups. A very significant step was made by V. Voevodsky in 1997, when he proved the Mi* *lnor conjecture [95] which asserts that if F is a field of characteristic 6= 2 , then KMi(F)=2KMi(F) ~=Hi* *et(F; Z=2) (see Definition 2.33). This fundamental theorem has many deep consequences. In particular, J. R* *ognes and C. Weibel were then able to use it in order to calculate the E2-term of the Bloch-Lichtenbaum * *spectral sequence Es;t2=) K-s-t(Q; Z=2) and, after a very tricky study of its differentials, to determine* * the groups Ki(Q; Z=2) . They could then deduce from the localization exact sequence the calculation of * *Ki(Z; Z=2) for all integers i . At that point, they were very lucky since all elements of the groups Ki(Z; * *Z=2) were detected by the elements of Ki(Z) which were already known by Theorem 9.2, Remark 9.3, Theorem * *9.10 and Corollary 9.11. Consequently, they obtained the following complete calculation of the 2-t* *orsion of the groups Ki(Z) . Theorem 9.18. (Voevodsky, Rognes-Weibel) K1(Z) ~=Z=2 and for i 2 , 8 >>>Z Z=2 finite odd torsion group, if i 1 mod 8, >>>Z=2 finite odd torsion group , if i 2 mod 8, >>< Z=16 finite odd torsion group , if i 3 mod 8, Ki(Z) ~=> >>>Z finite odd torsion group , if i 5 mod 8, >>>Z=(2(i + 1) ) finite odd torsion,groupif i 7 mod 8, >: 2 finite odd torsion group , otherwise. Proof. See [99], Table 1, and [77], Theorem 0.6. * * _|_| Remark 9.19. W. Browder had already observed in [36], Theorem 4.8, that the cyc* *lic factor of order 16 comes periodically in all groups Ki(Z) with i 3 mod 8 . This theorem has an immediate crucial topological consequence. Consider any pri* *me p 3 or5 mod 8 and the above map : (BGL(Z)+)^2! JK(Z; p)^2 which induces a split surjection on all homotopy groups. It is easy to check th* *at Theorem 9.18 shows that the homotopy groups of (BGL(Z)+)^2and of JK(Z; p)^2are the same. Therefore, The* *orem 9.17 implies that the induced homomorphism * : ssi((BGL(Z)+)^2) ! ssi(JK(Z; p)^2) is an isomorphism for all positive integers i . This and a similar argument for* * the map __ + : (BGL(Z[1_2]) )^2! J(p)^2 imply the following result. 48 Corollary 9.20. There are homotopy equivalences (BGL(Z)+)^2' JK(Z; p)^2 and (BGL(Z[1_2])+)^2' J(p)^2: Consequently, we may deduce the following theorem. Theorem 9.21. For any prime p 3 or5 mod 8 , one has the following commutative * *diagrams in which the rows are fibrations and where the right square is a pull-back square: _ _ U^2 ------! (BGL(Z[1_2])+)^2------!BO^2 ?? ? __ ? y ' ?yfp ?y_c _ U^2 ------! (BGL(Fp)+)^2 ---b---! BU^2 and SU^2 ------! (BGL(Z)+)^2 ------! BO^2 ?? ? ? y ' ?yfp ?yc SU^2 ------! (BSL(Fp)+)^2---b---! BSU^2: _ __ * *__ ' Here, the maps , fp, and fp are the compositions of the homotopy equivalence * * : (BGL(Z[1_2])+)^2-! _0 __ * * _ J(p)^2, respectively : (BGL(Z)+)^2'-!JK(Z; p)^2, with the maps , f0p,_0 and* * f0p. Moreover, and are the maps induced by the inclusions Z[1_2] ,! R and Z ,! R , and fp and fp * *are induced by the reduction mod p . Proof. This follows from the two diagrams introduced above and from_Corollary_9* *.20. A careful study of B"okstedt's construction implies the identification of the maps , , fp, fp. * * _|_| This result provides a very complete knowledge of the homotopy type of the K-th* *eory space BGL(Z)+ at the prime 2 . First of all, the homotopy groups Ki(Z) = ssi(BGL(Z)+) of this* * space are known (after 2-completion) by Theorem 9.18. Then, it is also possible to determine at the pr* *ime 2 the Hurewicz ho- momrphism hi : Ki(Z) ! Hi(GL(Z); Z) . If i 1 mod 4 and i 5 , then Ki(Z) ~=Z * *(finite group) by Theorem 9.1. Let us call bi a generator of the infinite cyclic summand of Ki* *(Z) . On the other hand, it follows from Theorem 9.5 that H*(GL(Z); Q) ~=Q(x5; x9; x13; : :x:4l+1; : :):* *, where the elements x4l+1 are primitive generators of degree 4l + 1 . Thus, for any integer i 1 mod 4 (w* *ith i 5 ) there exists a generator ai of an infinite cyclic summand of Hi(GL(Z); Z) with the property th* *at hi(bi) = iai+ (torsion element); where i is a positive integer. By using the fact that the Hurewicz homomorphism* * ssi(SU) ! Hi(SU; Z) acts in some sense as multiplication by (i-1_2)! (see [40], Theoreme 6) and the* * map : SU^2! (BGL(Z)+)^2 provided by Theorem 9.21, it is not difficult to compute the 2-primary part (i)* *2 of these integers i. 49 i - 1 Corollary 9.22. For all integers i 1 mod 4 with i 5 , (i)2= ((____2)!)2. Proof. See [21], Theoreme 4.17. * * _|_| Remark 9.23. In a similar way, one can compute the effect of the Hurewicz homom* *orphism on the 2-torsion classes of K*(Z) (see [21], Theoreme 4.22). In order to understand the homotopy type of BGL(Z)+ , one also needs to know it* *s Postnikov k-invariants ki+1(BGL(Z)+) 2 Hi+1(BGL(Z)+[i - 1]; Ki(Z)) (see Section 7). We know from Theo* *rem 7.17 that all k-invariants ki+1(BGL(Z)+) are cohomology classes of finite order. Definition 9.24. For any i 2 , let aei denote the order of the k-invariant ki* *+1(BGL(Z)+) in the cohomology group Hi+1(BGL(Z)+[i - 1]; Ki(Z)) . Corollary 9.25. For all integers i 2 , the 2-primary part (aei)2 of the intege* *r aei is given by 8 >>>((i_-_1)!), if i 1 mod 4, >>> 2 2 ><2 , if i 2 mod 8 and i 10, or if i = 3 or 7, (aei)2= > 16 , if i 3 mod 8 and i 11, or if i = 15, >>> >>>2(i + 1)2, if i 7 mod 8 and i 23, : 1 , otherwise. Proof. This follows directly from Proposition 7.10 and the description of the H* *urewicz homomorphism at the prime 2 provided by Corollary 9.22 and Remark 9.23 (see [21], Theoreme 5.15* * for more details). _|_| The fibration SU^2-! (BGL(Z)+)^2-! BO^2 given by Theorem 9.21 enables us to deduce two other important properties of th* *e algebraic K-theory space of Z . Let us first completely compute the 2-adic product structure of K*(Z) . Definition 9.26. The 2-adic product map in K*(Z) is the composition ? : Ki(Z) Kj(Z) -! Ki+j(Z) -! Ki+j(Z) Z^2; where the first arrow is the usual K-theoretical product defined in Definition * *4.5 and the second the tensor product of Ki+j(Z) with the inclusion of Z into the ring of 2-adic integers Z^2* *( i , j 1 ). We continue to denote this product by the symbol ? . Theorem 9.27. The 2-adic product ? : Ki(Z) Kj(Z) -! Ki+j(Z) Z^2 is trivial for all positive integers i and j , except if i j 1 mod 8 or i 1 * *mod 8 and j 2 mod 8 (or i 2 mod 8 and j 1 mod 8 ), where its image is cyclic of order 2 . In both* * exceptional cases the non-trivial element in the image of the 2-adic product map is the product of tw* *o elements of order 2 . Proof. The 2-adic product ? : Ki(Z) Kj(Z) -! Ki+j(Z) Z^2is clearly trivial fo* *r dimension reasons (see 50 Theorem 9.18) whenever i and j do not belong to one of the following six cases: i 1 mod 8and j 1 mod 8 ; i 1 mod 8and j 2 mod 8 ; i 2 mod 8and j 5 mod 8 ; i 2 mod 8and j 7 mod 8 ; i 3 mod 8and j 7 mod 8 ; i 5 mod 8and j 5 mod 8 : In order to compute these six products, let us consider the following commutati* *ve diagram induced by the inclusion Z ,! R : Ki(Z) Kj(Z) --**----!ssiBO ssjBO ?? ? y ? ?y ssi+jSU Z^2--*----! Ki+j(Z) Z^2---*---! ssi+jBO Z^2; where the right vertical arrow is the composition of the product map in ss*BO w* *ith the tensor product with Z^2and where the bottom sequence is the homotopy exact sequence of the top* * fibration of the second diagram in Theorem 9.21. If i + j is even, * is injective since ssi+jSU = 0 . C* *onsequently, this diagram detects the 2-adic product Ki(Z) ? Kj(Z) when i + j is even. This produces the * *calculation of the product in three of the above six cases. The 2-adic product turns out to be trivial in * *the last two cases. For the case i j 1 mod 8 , recall that Ki(Z) ~=Z Z=2 (finite odd torsionigroup)f i 1 mo* *d 8 (and i 9 ) and that Ki(Z) ~=Z=2 (finite odd torsion group)if i 2 mod 8 (see Theorem 9.18). L* *et us denote by yi, respectively by zi, the element of order 2 in Ki(Z) when i 1 mod 8 , respectiv* *ely when i 2 mod 8 . Our argument shows that if i j 1 mod 8 , the 2-adic product ? : Ki(Z) Kj(Z) * *! Ki+j(Z) Z^2 satisfies yi? yj= zi+j (where zi+j is also written for the image of the element zi+j of Ki+j(Z) under * *the homomorphism Ki+j(Z) ! Ki+j(Z) Z^2) and vanishes on other elements. In particular, we get t* *he isomorphism Ki(Z) Z^2~=(K1(Z) ? Ki-1(Z)) Z^2; for i 2 mod 8 , which is useful in order to calculate the 2-adic product in th* *e three remaining cases of the above list (see [14], Theorems 5.6, 5.7, 5.8 and 5.9 for the details of all the* *se computations). _|_| Finally, the fibration SU^2-! (BGL(Z)+)^2-! BO^2 also provides the determination of the Hopf algebra structure of the mod 2 coho* *mology of the infinite general linear group GL(Z) as a module over the Steenrod algebra. Theorem 9.28. There is an isomorphism of Hopf algebras and of modules over the * *Steenrod algebra H*(GL(Z); Z=2) ~=H*(BGL(Z)+; Z=2)~=H*(BO; Z=2) H*(SU; Z=2) ~=Z=2[w1; w2; w3; : :]: Z=2(u3; u5; u7; * *: :):; where the wj's are the Stiefel-Whitney classes of degree j ( j 1 ) and the cla* *sses u2k-1 are exterior classes of degree 2k - 1 ( k 2 ). Proof. The recent paper [15] contains the proof of this assertion (see Theorem * *1 of [15]) and an explicit def- inition of the exterior classes u2k-12 H2k-1(BGL(Z)+; Z=2) (see [15], Definitio* *n 10 and Remark 14). The 51 classes wj are the images under * : H*(BO; Z=2) ! H*(BGL(Z)+; Z=2) of the unive* *rsal Stiefel-Whitney classes in H*(BO; Z=2) (see [66], Chapter 7). Notice that an additive version o* *f the above isomorphism has been conjectured in [42], Corollary 4.3, and that the statement of the theo* *rem can also be deduced from Theorem 4.3 and Remark 4.5 of [68] together with Corollary 9.20. * * _|_| All these results provide a very deep knowledge of the homotopy type of the K-t* *heory space BGL(Z)+ at the prime 2 . At odd primes, the situation is more difficult. If we would like * *to understand BGL(Z)+ at an odd prime l , we still have B"okstedt's space JK(Z; p) for all primes p . It* * is even possible to prove that if p is well chosen, i.e., if p generates the multiplicative group (Z=l2)*, the* *n the l-completion JK(Z; p)^l of JK(Z; p) does not depend on p (see [21], Proposition 3.24): we shall denote * *it by JKZ^l. Definition 9.29. An odd prime l is called a Vandiver prime if it does not divid* *e the class number of the maximal real subfield of the cyclotomic field Q(l) , where l denotes an l-th pr* *imitive root of unity. It is known that all odd primes 400000000 are Vandiver primes and it is a conjecture* * that all odd primes are Vandiver primes (see [98], Section 8.3). Theorem 9.30. For any Vandiver prime l , the space JKZ^lis a direct factor of (* *BGL(Z)+)^l. Proof. This theorem is due to C. Ausoni (see [21], Theoreme 3.47) and its proo* *f is based on the result by W.G. Dwyer and S.A. Mitchell which asserts that (U=O)^lis a retract of (BGL(* *Z[1_l])+)^lfor Vandiver primes l (see [43], Example 12.2). * * _|_| Again, it is easy to compute the homotopy groups of JKZ^l: they contain the ele* *ments of infinite order given by Theorem 9.1 and the torsion classes detected by D. Quillen using the K* *-theory of finite fields (see Theorem 9.10). Remark 9.31. If l is a regular prime, the Lichtenbaum-Quillen conjecture (see R* *emark 9.16) is equivalent to the conjecture saying that there is a homotopy equivalence (BGL(Z)+)^l' JKZ^* *l. However, at irregular primes, this cannot be true since the even dimensional homotopy groups of JKZ^l* *do not contain the irregular torsion discovered by C. Soule (see Theorem 9.13). Nevertheless, Theorem 9.30 helps us to understand the homotopy type of BGL(Z)+ * *since it implies the three following results (see [21], Theoremes 4.17 and 5.14 and Proposition 5.18* *). Corollary 9.32. For any integer i 1 mod 4 ( i 5 ), the l-primary part (i)l of* * the integer i (see Corollary 9.22) which describes the effect of the Hurewicz homomorphism on the * *elements of infinite order bi2 Ki(Z) has the following property: if l is a Vandiver prime, then (i)l= ((i_* *-_12)!)l. Corollary 9.33. For any integer i 1 mod 4 ( i 5 ), the l-primary part (aei)l * *of the order aei of the k-invariant ki+1(BGL(Z)+) satisfies: if l is a Vandiver prime, then (aei)l ((i_* *-_12)!)l. Corollary 9.34. If l is a regular prime and if the Lichtenbaum-Quillen conjectu* *re is true at l (see Remark 9.31), then the integers (aei)l can be exactly determined: 8 >><((i_-_1)!)l, if i 1 mod 4, (aei)l= > 2 >:l(i + 1)l, if 2(l - 1) is a proper divisor of (i + 1* *), 1 , otherwise, except for (ae11)3 which is equal to 3 . 52 10. Further developments The goal of this paper was to show how some topological methods can provide ver* *y general and deep results on the algebraic K-theory of rings. We especially emphasized the use of the inf* *inite loop space structure of the K-theory space BGL(R)+ of any ring R , of cohomological calculations for* * linear groups, of the relationships between K-theory and linear group homology, and of the study of h* *omotopical approximations. Of course, the arguments presented here do not represent all the topological co* *nsiderations which can give interesting K-theoretical information. It is not our purpose to describe these other ideas in details in this paper, b* *ut we just want to mention some of them. W. Browder applied in [36] the techniques of homotopy theory with finite * *coefficients (see [69]) in order to investigate the algebraic K-groups with coefficients in Fp for any prime p a* *nd to deduce nice theorems on the ordinary algebraic K-theory: in particular, he could exhibit a periodici* *ty result for the groups Ki(Z) (see Theorem 9.10, Theorem 9.18 and Remark 9.19). F. Waldhausen introduced the * *S-construction which enabled him to present K-theoretical notions and results in a very general way * *over suitable categories (see for instance [97], Section 1.3, and [62]). W.G. Dwyer and E.M. Friedlander* * constructed in [41] another spectrum associated with a ring R , theetale K-theory spectrum of R , whose hom* *otopy groups are called theetale K-groups of R : they are in principle easier to calculate and some str* *ong results are known on their relationships with the ordinary algebraic K-groups of R ; however, the ho* *momorphism relating these two K-theories is still the object of several difficult conjectures (see [41] a* *nd [42] for example). Some other impressive progress has been made by using techniques from stable homotopy theo* *ry (see for instance the works by M. B"okstedt, W.G. Dwyer, R. McCarthy, S.A. Mitchell, J. Rognes and C.* * Weibel). 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