Algebraic K-theory of rings from a topological viewpoint

Because of its strong interaction with almost every part of pure mathematics, algebraic K-theory has had a spectacular development since its 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 theorems. Since the algebraic K-groups of a ring R are the homotopy groups of a topological space associated with the general linear group over R, it is obvious that many general results follow from arguments from homotopy theory. This paper is essentially devoted to some of them: it explains in particular how methods from stable homotopy theory, group cohomology and Postnikov theory can be used in algebraic K-theory.

. The product structure in algebraic K-theory and the K-theory spectrum 24 5. The algebraic K-theory of finite fields 30 6. The Hurewicz homomorphism in algebraic K-theory 37 7. The Postnikov invariants in algebraic K-theory 46 8. The algebraic K-theory of number fields and rings of integers 57 9. The algebraic K-theory of the ring of integers Z 63 10. Further developments 77 References 78

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 preoccupations of the first K-theorists because of its relationships with linear groups which play a crucial role in almost all subjects in mathematics. Later, J. Milnor and D. Quillen introduced a very general notion of algebraic K-groups K i (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 over rings!
The abelian groups K i (R) are homotopy groups of a space which is canonically associated 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 introduce the definition of the groups K i (R) for all integers i ≥ 0 (in Sections 1, 2 and 3), to explore their structure (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, instead of looking at the category of finitely generated projective modules over a ring, we apply the same ideas to other categories, we also get interesting and important developments of algebraic K-theory in various directions 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]).
Throughout the paper, all maps between topological spaces are supposed to be continuous pointed maps and all rings are rings with units.

The origins of algebraic K-theory
The very beginning of algebraic K-theory is certainly due to some general considerations 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 an 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 denoting this group of classes (K = Klassen) of objects of C. This group K(C) was the first algebraic K-group: it is now called the Grothendieck group of the category C.
Let us explain in more details the definition of a Grothendieck group by looking at two classical examples. First, let R be a ring and P(R) the category of finitely generated projective R-modules (see [  Thus, every element of K 0 (R) can be written as a difference [P ] − [Q] of two generators 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 K 0 (R) if and only if there is a free R-module R s such that P ⊕ R s ∼ = Q ⊕ R s (in that case, the R-modules P and Q are called stably isomorphic).
A homomorphism of rings ϕ : R → R induces a homomorphism of abelian groups ϕ * : K 0 (R) −→ K 0 (R ) 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 ∼ = R n and consequently, (R ⊗ R P ) ⊕ (R ⊗ R Q) ∼ = R ⊗ R R n ∼ = (R ) n : in other words R ⊗ R P is a finitely generated projective R -module. Therefore, let us define for all [P ] ∈ K 0 (R) ϕ * ([P ]) = [R ⊗ R P ] ∈ K 0 (R ); K 0 (−) 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 K 0 (R) are the following two questions 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 questions would have a positive answer, then the group K 0 (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. 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 rings of algebraic integers in number fields. The second classical example of a Grothendieck group is given by the topological K-theory which was introduced by A. Grothendieck in 1957 (see [33]) and developed by M. F. Atiyah and 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. where ⊕ is written for the Whitney sum of vector bundles. The abelian group K 0 F (X) is called the topological K-theory of X (see [19] or Part II of [51] for more details). with f * (p): f * (E) → X given by f * (p)(x, e) = x defines an F-vector bundle over X. Thus, the map f induces a homomorphism of abelian groups f * : K 0 F (Y ) −→ K 0 F (X) and it turns out that K 0 F (−) is a contravariant functor from the category of compact Hausdorff topological spaces to the category of abelian groups.
Remark 1.5. It is not difficult to show that the group K 0 F (X) splits as K 0 F (X) ∼ = Z ⊕ K 0 F (X), where K 0 F (X), the reduced topological K-theory of X, is the kernel of the homomorphism K 0 F (X) → Z which associates to each vector bundle its rank.
For example, the calculation of K 0 F (X) in the case where X is an n-dimensional sphere S n is provided by the celebrated Theorem 1.6 (Bott periodicity theorem).
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 or C, and R(X) the ring of continuous functions X → F. There is an isomorphism of abelian groups K 0 F (X) ∼ = K 0 (R(X)).
Remark 1.8. It is also possible to describe K 0 (R) in terms of idempotent matrices 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 in the next sections.

The functors K 1 and K 2
One of the main objects of interest in linear algebra over a ring R is the general linear group GL n (R) consisting of the multiplicative group of n × n invertible matrices with coefficients in R. In order to look at all invertible matrices of any size in the same group, observe that GL n (R) may be viewed as a subgroup of GL n+1 (R) via the upper left inclusion A → ( A 0 0 1 ) and consider the direct limit which is called the infinite general linear group. Algebraic K-theory is essentially the study of that group for any ring R. To begin with, let us investigate the commutator subgroup of GL(R).
Definition 2.1. Let R be any ring, n a positive integer, i and j two integers with 1 ≤ i, j ≤ n, i = j, and λ an element of R; let us define the matrix e λ i,j to be the n × n matrix with 1's on the diagonal, λ in the (i, j)-slot and 0's elsewhere: such a matrix is called an elementary matrix in GL n (R). Let E n (R) denote the subgroup of GL n (R) generated by these matrices and let 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 [  (a) Any triangular matrix with 1's on the diagonal and coefficients in R belongs to the group E(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
A ring homomorphism f : R → R induces obviously a homomorphism of abelian groups f * : K 1 (R) → K 1 (R ) and K 1 (−) is a covariant functor from the category of rings to the category of abelian groups.
Remember that the abelianization of any group G is isomorphic to its first homology group with integral coefficients H 1 (G; Z) (see [37,Section II.3] or [50,Section VI.4]).
If R is commutative, the determinant of square matrices is defined and we may consider the group SL n (R) of n × n invertible matrices with coefficients in R and determinant +1, and the infinite special linear group This provides the extension of groups is clearly a subgroup of SL(R) since any elementary matrix e λ i,j has determinant +1. By looking at the above extension and taking the quotient by E(R), one gets the short exact sequence of abelian groups which splits since the composition of the inclusion of R × = GL 1 (R) = GL 1 (R)/E 1 (R) into the group GL(R)/E(R) = K 1 (R) with the surjection K 1 (R) → R × is the identity. Therefore, we can introduce the functor SK 1 (−) and obtain the next theorem. Definition 2.8. For any commutative ring R, let Theorem 2.9. For any commutative ring R, Consequently, it is sufficient to calculate SK 1 (R) in order to understand K 1 (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 euclidean ring or the ring of integers in a number field, then SK 1 (R) = 0 and the determinant induces an isomorphism Proof: See [78, Sections 2.2 and 2.3]. If R is a Dedekind domain, it is possible to prove that K 1 (R) is generated by the image of GL 2 (R) in K 1 (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 SK 1 (R) consists of Mennicke symbols. Remark 2.14. In that case, SK 1 (R) is in general non-trivial. However, if R is a Dedekind domain such that R/m is a finite field for each nontrivial maximal ideal m of R, then SK 1 (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. Milnor and M. Kervaire in the late sixties when they investigated the universal central extension of the group E(R) (see [65] and [54] 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.
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 A ring homomorphism f : R → R induces group homomorphisms St(R) → St(R ) and E(R) → E(R ) and consequently a homomorphism of abelian groups f * : K 2 (R) → K 2 (R ). 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 K 2 (R) measures the non-obvious relations of E(R). From that viewpoint, the knowledge of K 2 (R) is essential for the understanding of the structure of the group E(R).
Remark 2.18 immediately implies the following homological interpretation of the functor K 2 (−).

Corollary 2.25. For any ring
Remark 2.26. The definitions of K 1 (R) and K 2 (R) show the existence of the exact sequence where the middle arrow is the composition of the homomorphism ϕ : For the remainder of this section, let us assume that R is a commutative ring. Consider the above homomorphism ϕ : St(R) E(R). If x and y belong to E(R), one can choose elements X and Y in St(R) such that ϕ(X) = x and ϕ(Y ) = y. Of course, X and Y are not unique; however, the commutator [X, Y ] is uniquely determined by x and y, because for any a and b in ker ϕ one has [Xa,  Many results have been obtained on K 2 (R) in the case where R is a field. The first one asserts that K 2 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 suggests a possible algebraic generalization of the functor K 2 (−) 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 F × is where F × is considered as an abelian group and In this algebra, one can consider the ideal I generated by all elements of the form u ⊗ (1 − u) when u belongs to F × . Then the Milnor K-theory of F is defined by For each positive integer n, the elements of K M n (F ) are the symbols {u 1 , u 2 , . . . , u n }, with the u i 's in F × satisfying the following rules (additively written): and K M * (F ) has an obvious multiplicative structure given by the juxtaposition of symbols.
A lot of work has been done in the study and calculation of Milnor K-theory. However, we shall not discuss it here since the purpose of this paper is to study another generalization of K 1 and K 2 , based on topological considerations: Quillen's higher K-theory.

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 discuss 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 fundamental group π 1 (X) and with any discrete group G its classifying space BG which is an Eilenberg-MacLane space K(G, 1). This means that the homotopy groups of BG are all trivial except for π 1 (BG) ∼ = G and implies 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 → π 1 (X) fulfills π 1 (BG) ∼ = G, but it is in general not true that the classifying space of the fundamental group of a space X is homotopy equivalent to X: in other words, a topological space X is in general 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 (see [70], [ Theorem 3.1. Let X be a connected CW-complex, P a perfect normal subgroup of its fundamental group π 1 (X). There exists a CW-complex X + P , obtained from X by attaching 2-cells and 3-cells, such that the inclusion i : X → X + P satisfies the following properties: (a) the induced homomorphism i * : π 1 (X) → π 1 (X + P ) is exactly the quotient map π 1 (X) for any local coefficient system A on X + P , (c) the space X + P is universal in the following sense: if Y is any CW-complex and f : X → Y any map such that the induced homomorphism f * : π 1 (X) → π 1 (Y ) fulfills f * (P ) = 0, then there is a unique map f + : X + P → Y such that f + i = f . In particular, X + P is unique up to homotopy 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 π 1 (X); then, we build X + P by attaching 3-cells to the space we just obtained because X + P must have the same homology as the original space X. Observe that this creates a lot of new elements in the homotopy groups of X The universal property of the plus construction implies the following assertion.

Corollary 3.2.
Let X and X be two connected CW-complexes, P and P two perfect normal subgroups of π 1 (X) and π 1 (X ) respectively and f : X → X a map such that f * (P ) ⊂ P . Then there is a map f + : X + P → X + P (unique up to homotopy) making the following diagram commutative: and it is easy to check that the plus construction is functorial.
Let us also mention the following important property.

Lemma 3.3.
If X is the covering space of X associated with the perfect normal subgroup P of π 1 (X), then there is a homotopy equivalence ( X) + P X + P between ( X) + P and the universal cover X + P of X + P . Proof: See [59, Proposition 1.1.4] for more details.
Remark 3.4. If P is the maximal perfect normal subgroup of π 1 (X), it is usual 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 + P since the 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 group G X and a map t X : BG X = K(G X , 1) → X which is natural with respect to X and has the following properties: (a) the homomorphism (t X ) * : This implies the following consequences. Corollary 3.6. For every connected CW-complex X, the kernel P X of (t X ) * : G X π 1 (X) is perfect.
in which X is the universal cover of X. Both horizontal maps have the same homotopy fiber π 1 (X) and t X induces an isomorphism on homology with any local coefficient system. Therefore, the comparison theorem for the Serre spectral sequences of both horizontal maps implies that ( t X ) * : H 1 (Y ; Z) → H 1 ( X; Z) is an isomorphism and that H 1 (Y ; Z) vanishes since H 1 ( X; Z) = 0. The homotopy exact sequence of the fibration shows that π 1 (Y ) ∼ = P X . Consequently, (P X ) ab ∼ = π 1 (Y ) ab ∼ = H 1 (Y ; Z) = 0, in other words, P X is a perfect group.
Theorem 3.7. For every connected CW-complex X, there exists a group G X together with a perfect normal subgroup P X such that one has a homotopy equivalence (BG X ) + P X

X.
Proof: For a connected CW-complex X, let us consider the group G X , the map t X : BG X → X and the perfect group P X = ker((t X ) * : G X π 1 (X)) given by Theorem 3.5 and Corollary 3.6. Then, consider the plus construction (BG X ) + P X and apply Theorem 3.1 (c): there is a map t + X : (BG X ) + P X → X which induces an isomorphism on π 1 and on all homology groups. The generalized Whitehead theorem (see [48, Corollary 1.5], or [29,Proposition 4.15]) finally implies that t + X is a homotopy equivalence. between the corresponding CW-complexes which induces an isomorphism on the fundamental group and on all homology groups. Again, it follows from the generalized Whitehead theorem that Bf + : BG + P → BG + P is a homotopy equivalence. Consequently, this establishes a very nice one-to-one correspondence between group theory and topology (see [28,Section 11], and [49, Section 2]): equivalence classes of topogenic groups ←→ homotopy types of CW-complexes Remark 3.9. By Theorem 3.1 the space BG + P associated with the topogenic group (G, P ) satisfies the following properties: (a) π 1 (BG + Example 3.10. The perfect groups correspond to the simply connected CW-complexes, because for any perfect group P , the space BP + P associated with the topogenic group (P, P ) has trivial fundamental group π 1 BP + P ∼ = P/P = 0. Example 3.11. Let Σ ∞ = lim − → n Σ n be the infinite symmetric group and A ∞ = lim − → n A n the infinite alternating group which is perfect. The Barratt-Priddy theorem [26] asserts that the topogenic group (Σ ∞ , A ∞ ) corresponds to (BΣ ∞ ) + A∞ which is homotopy equivalent to the connected component of the space Q 0 S 0 = lim − → n Ω n S n whose homotopy groups are the stable homotopy groups of spheres In order to define a suitable generalization of the functor K 1 (−) and K 2 (−) 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 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 Let us check that this definition extends the definition of K 1 (R) and K 2 (R) given in Section 2. Remark 3.9 (a) shows that π 1 (BGL(R) + ) ∼ = GL(R)/E(R) and this group is exactly K 1 (R) according to Definition 2.6.
Notice also that Lemma 3.3 shows that the universal cover of BGL(R) + is the space 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 integers i ≥ 2, In particular, the group K 2 (R) given by Definition 3.12 coincides with the group π 2 (BE(R) + ) which is isomorphic to H 2 (BE(R) + ; Z) ∼ = H 2 (E(R); Z) (see Remark 3.9 (b)) because of the Hurewicz theorem. Corollary 2.25 then implies that it is isomorphic to K 2 (R) as defined in Definition 2.23.
Of course, a ring homomorphism f : R → R induces a group homomorphism GL(R) → GL(R ) whose restriction to E(R) sends E(R) into E(R ). Thus, Corollary 3.2 implies the existence of a map BGL(R) + → BGL(R ) + and consequently of a homomorphism of abelian groups 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): In the same way, we can prove the following result.
Proof: Let us consider the universal central extension and the associated fibration of classifying spaces Since E(R) and St(R) are perfect, one can perform the + construction to both spaces BSt(R) and BE(R). If one denotes by F the homotopy fiber of the induced map one has the following commutative diagram where both rows are fibrations: is the center of St(R) by Theorem 2.22, the action of π 1 (BE(R)) on the homology of BK 2 (R) is trivial. The same holds for the second fibration since BE(R) + is simply 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 * (BK 2 (R); Z) → H * (F ; Z) is an isomorphism. Since St(R) is perfect, the space BSt(R) + is also simply connected. On the other hand, it is known that H 2 (BSt(R) + ; Z) ∼ = H 2 (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 Since π 1 (F ) ∼ = K 2 (R), it is an abelian group. Consequently, the Hurewicz homomorphism π 1 (F ) → H 1 (F ; Z) is an isomorphism. Then, consider the commutative diagram where the vertical arrows are Hurewicz homomorphisms. The left vertical arrow is an isomorphism since The homotopy exact sequence of that fibration shows that for all i ≥ 3 and that BSt(R) + is the 2-connected cover of BGL(R) + . Finally, it follows from the Hurewicz theorem that , . . . whose homology represents the K-groups of R. Unfortunately, we do not have any explicit description of the groups G m (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 In the present paper we want to concentrate our attention on the first definition 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 K i (R) and by its Postnikov k-invariants (see Section 7). The space BGL(R) + has actually many other interesting properties. The remainder of the paper is devoted to the investigation of some of them.

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) + of any ring R has actually a very rich structure. Let us start by considering a ring R and the homomorphism given by which endows BGL(R) + with the following structure. Now, let R and R be two rings and let us denote by R ⊗ R the tensor product R ⊗ Z R over Z. The tensor product of matrices induces a homomorphism

By composing this map with the map induced by the upper left inclu
, where x 0 and y 0 are the base points of BGL m (R) + and BGL n (R ) + respectively, and where "−" is the subtraction in the sense of the H-space structure of the space BGL(R ⊗ R ) + . Since the maps γ R,R m,n are compatible (up to homotopy) with the stabilizations i R,R m,n : which is unique up to weak homotopy (see [59, Lemme 2.1.6 and Remarque 2.1.9]). By definition, this map γ R,R is homotopic to the trivial map on the wedge BGL(R) + ∨ BGL(R ) + . Consequently, it finally induces a map It turns out that this map γ R,R is natural in R and R , bilinear, associative and commutative, up to weak homotopy (see [ (Loday). For all rings R and R , and for all integers i, j ≥ 1, the product map One can then immediately deduce the following properties (see [59, Théorème 2.1.11]).

Proposition 4.3. The product map
is natural in R and R , 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 K i (R) ⊗ K j (R ) and we shall also denote it by the symbol : Now, let us look at the special case where R = R. If R is a commutative ring, the ring homomorphism ∇ : R ⊗ R → R given by ∇(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 denoted by ) This product satisfies: The remainder of this section is devoted to further investigation of the H-space structure of the space BGL(R) + (see [59,Sections 1.4 and 2.3]). Let us first consider the ring of integers R = Z. Its cone CZ is the set of all infinite matrices with integral coefficients having only a finite number of non-trivial elements on each row and on each column. This set turns out to be a ring by the usual addition and multiplication of matrices. Let JZ be the ideal of CZ which consists of all matrices having only finitely many non-trivial coefficients. Finally, let us define the suspension of Z to be the quotient ring ΣZ = CZ/JZ. Definition 4.7. For any ring R, the suspension of R is the ring This element is invertible since τ τ t = 1 in ΣZ and consequently τ ∈ GL 1 (ΣZ). Let [P ] be a generator of the group K 0 (R), where P is a finitely generated projective R-module. There is an R-module Q such that P ⊕Q ∼ = R n for some n and an R-module homomorphism R n → R n which is the identity on P and trivial on Q. Let us call p the n×n matrix with coefficients in R corresponding to that homomorphism. By using the tensor product of matrices, one can construct the element τ ⊗ p + 1 ⊗ (1 − p), which is an invertible n × n matrix with coefficients in ΣR, in other words, which belongs to GL(ΣR). This produces a homomorphism This fact is proved in [53] and can be generalized. Let σ : Z → GL 1 (ΣZ) be the group homomorphism given by σ(1) = τ . It induces a map σ + : S 1 BZ + → BGL(ΣZ) + and we write ε R for the composition For any generator [P ] of K 0 (R), let us choose a representative ζ P : The point is that the homotopy type of the space BGL(ΣR) + depends very strongly on the homotopy type of BGL(R) + because of the following result.
Theorem 4.9. The map ε R is a natural homotopy equivalence: Proof: By definition, 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: There is a natural homotopy equivalence Remark 4.12. Of course, Theorem 4.9 shows that for any ring R, the spa- This enables us to define an Ω-spectrum whose 0-th space is the space Definition 4.13. For any ring R, the K-theory spectrum of R is the Ω-spectrum K R whose n-th space is ( We shall also use the 0-connected cover X R of the K-theory spectrum K R of a ring R. Definition 4.14. For any ring R, the 0-connected K-theory spectrum of R is the Ω-spectrum X R whose n-th space is (X R ) n = BGL(Σ n R) + (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 π i (X(n)) ∼ = π i (X) for all i ≥ n.
Of course, since these spectra are Ω-spectra, their homotopy groups are for any i ∈ Z; in particular, they are in general non-trivial if i < 0. On the other hand, for X R , and we may conclude that if i ≥ 1. Therefore, Corollary 4.10 implies the following result for i ≥ 0.
Remark 4.17. Since the K-groups of a ring R in positive dimensions are the homotopy groups of its 0-connected K-theory spectrum X R (or of K R ), they are very strongly related to the homology groups of X R via the stable Hurewicz homomorphism, as we shall see in Section 6. Therefore, it would be extremely useful to be able to compute In a recent paper [63] which generalizes MacLane's Q-construction for computing the stable homology of Eilenberg-Maclane spaces [60], R. Mc-Carthy obtains an explicit chain complex whose homology is the homology of X R . This promising idea should provide more information on H i (X R ; Z) and consequently on the algebraic K-groups K i (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 R , together with their associated 0-connected spectra X R and X R , and let S denote the sphere spectrum. The external product is defined as follows. If x ∈ π i (X R ) and y ∈ π j (X R ) are represented by maps of spectra α : S → X R of degree i and β : [94, p. 270]).
which was the key ingredient in Definition 4.2 extends of course to a map for all n and m and consequently to a pairing of spectra Moreover, if R is commutative, the ring structure of K * (R) is given by for all i, j ≥ 1.

The algebraic K-theory of finite fields
When D. Quillen introduced the higher algebraic K-groups, one of his great achievements was to completely compute them for finite fields (see [71]). Let p be a prime, q a power of p and let F q denote the field with q elements. Since F q is a field, it is known by Theorem 1.2 that In order to calculate K i (F q ) for any positive integer i, Quillen's brilliant idea was to construct a topological model for the space BGL(F q ) + using well known spaces. He considered the classifying space BU of the infinite unitary group U and the Adams operation Ψ q : BU → BU . Remember that for i ≥ 1 π i (BU ) = 0 if i is odd and π i (BU ) ∼ = Z if i is even (by Bott periodicity, see [35] or Chapter 10 of [51]), and that the homomorphism Ψ q * : π 2j (BU ) → π 2j (BU ) induced by Ψ q is multiplication by q j (see [1,Corollary 5.2]).
where BU [0,1] is the path space of BU and ∆ the map sending a path in BU to its endpoints. A point of F Ψ q is a pair (x, u), where x is a point of BU and u a path in BU joining Ψ q (x) to x. In other words, F Ψ q is the homotopy theoretical fixpoint set of Ψ q . According to Lemma 1 of [71], it turns out that if d : Proposition 5.2. For any integer q ≥ 2, the space F Ψ q is simple and its homotopy groups are Proof: Let us consider the fibration Since the action of π 1 (F Ψ q ) on the higher homotopy groups π i (F Ψ q ) comes from the action of π 1 (BU ) on π i (F Ψ q ), it is trivial because BU is simply connected. The calculation of π i (F Ψ q ) directly follows from the homotopy exact sequence of the above fibration 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 = p. Quillen's main argument is based on the calculation of the cohomology of the space F Ψ q and of the infinite general linear group GL(F q ). Let us start by defining some classes in the cohomology of F Ψ q . It is well known that where the c i 's and the c i 's are the integral universal Chern classes, respectively the mod l universal Chern classes, of degree 2i (see [66,Chapter 14]).
and the i-th mod l Chern class of F Ψ q is where ϕ * : H * (BU ; A) → H * (F Ψ q ; A) is the homomorphism induced by ϕ, for A = Z and A = Z/l respectively.
The diagram occuring in Definition 5.1 induces the following commutative diagram for any abelian group A in which the columns are exact sequences of pairs: In this diagram, γ and δ are injective because where the horizontal homomorphisms are induced by the reduction mod(q i − 1). It follows from the commutativity of the diagram that . This element is related to the integral Chern class c i ∈ H 2i (F Ψ q ; Z) by the formula induced by the obvious surjection Z/(q jr − 1) Z/l.
By using the Eilenberg-Moore spectral sequence of the fibration Quillen was able to calculate the cohomology of F Ψ q with coefficients in Z/l.

Theorem 5.6.
(a) The monomials c α1 r c α2 2r c α3 3r · · · e β1 r e β2 2r e β3 3r · · · , with α j ≥ 0 and β j = 0 or 1, form an additive basis for H * (F Ψ q ; Z/l). 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 ρ : G → GL n (F q ) a representation of G over the field F q with q elements. Let us denote by ρ the representation ρ = ρ ⊗ Fq F q of G over the algebraic closure F q of F q . We can look at the complex valued function on G defined by χ ρ (g) = ι(λ k (g)) for g ∈ G, where ι is an embedding F * q → C * and {λ k (g)} is the set of eigenvalues of ρ(g). It turns out that χ ρ is the character of a unique virtual complex representation ρ of G; therefore, χ ρ belongs to the complex representation ring R(G) = R C (G) of G and we get a homomorphism R Fq (G) → R(G) which maps the class of the character of ρ to χ ρ . In fact, ρ is stable under the Adams operation Ψ q (see [71,Section 7]) and the previous homomorphism is actually R Fq (G) → R(G) Ψ q . If we compose it with the classifying map R(G) → [BG, BU ] sending a complex representation to the corresponding homotopy class of maps between classifying spaces, we obtain the homomorphism On the other hand, observe again the fibration F Ψ q ϕ −→ BU Ψ q −1 −→ BU and remember that a point of F Ψ q is a pair (x, u), where x is a point of BU and u a path joining Ψ q (x) to x: this implies that for any Y , a map Y → F Ψ q can be identified with a pair consisting of a map f : Y → BU together with a homotopy joining Ψ q f to f . Consequently, ϕ induces a homomorphism which is clearly surjective. By looking at the fibration obtained by looping the base space of the above fibration, one gets that ϕ * is an isomorphism if [Y, U ] = 0. Therefore, if [BG, U ] = 0, the above homomorphism τ can be viewed as a homomorphism This is the case for G = GL n (F q ) and for the direct limit G = GL(F q ) = lim − → n GL n (F q ) according to Lemma 14 of [71].
Definition 5.7. Let G = GL n (F q ) and ρ =id: GL n (F q ) → GL n (F q ). The Brauer lift is the homotopy class of maps b n = τ (id) ∈ [BGL n (F q ), F Ψ q ]. By passing to the direct limit GL(F q ) = lim − → n GL n (F q ), one obtains a homotopy class of maps For simplicity, we shall also denote by b ∈ [BGL(F q ), BU] the composition of this last homotopy class of maps with the inclusion ϕ : F Ψ q → 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 Proof: (See [71, Theorems 2, 3, 4, 5, 6 and 7] for the details.) Let p be a prime, q a power of p and r be as in Definition 5.5. The argument is based on the investigation of the map 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 mod l homology of F Ψ q for any prime l = p. On the other hand, using techniques from homology theory of finite groups, it is possible to calculate H * (GL n (F q ); Z/l) for all positive integers n and consequently H * (GL(F q ); Z/l) ∼ = lim − → n H * (GL n (F q ); Z/l), and to prove that the homomorphism (b + ) * : This holds if b + induces an isomorphism on homology with coefficients in Q, in Z/p and in Z/l for all primes l = p. This is already done for coefficients in Z/l. It is easy to check that H i (BGL(F q ) + ; Q) ∼ = H i (GL(F q );Q)= 0 for all i ≥ 1 because H i (GL(F q ); Q) ∼ = lim − → n H i (GL n (F q ); Q) = 0 since GL n (F q ) is a finite group. On the other hand, we know from Proposition 5.2 that the homotopy groups of F Ψ q are torsion groups which are p-torsion free. Thus, by Serre class theory (see [82, Chapitre I]), all integral homology groups of F Ψ q are also torsion groups which are p-torsion free: in other words, H i (F Ψ q ; Q) = 0 and H i (F Ψ q ; Z/p) = 0 for all i ≥ 1. Thus, b + : BGL(F q ) + → F Ψ q is a homotopy equivalence and we get the statement of the theorem. This result is important because it provides a convenient topological model F Ψ q for the algebraic K-theory space BGL(F q ) + . In particular, an immediate consequence of it is the calculation of the algebraic K-groups of all finite fields: Proposition 5.2 and Theorem 5.8 imply the following result (see [71,Theorem 8]). Corollary 5.9. For any prime power q, the algebraic K-theory of the finite field F q is given by This result was the first determination of K-groups and initiated in some sense the research in the algebraic K-theory of rings.

The Hurewicz homomorphism in algebraic K-theory
The computation of the algebraic K-groups of finite fields was the first impressive K-theoretical result. It turns out that it is actually difficult to perform many other computations. However, this is not a surprise because the algebraic K-groups are homotopy groups and it is never easy to compute 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 actually 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 infinite Steinberg group St(R). They are exhibited by the Hurewicz homomorphisms Of course, since BGL(R) + , BE(R) + and BSt(R) + are connected, simply connected and 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, The general objective of this section is to approximate the size of the kernel and of the cokernel of h i in higher dimensions. We will proceed from different points of view (see also [7], [9], [12] and [14]).
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 homomorphismh i : π i (X) −→ H i (X; Z), defined for all integers i, which fits into the long stable Whitehead exact sequence. This sequence can be defined as follows. Consider the sphere spectrum S. It is (−1)-connected with π 0 (S) ∼ = Z and if we kill all its homotopy groups in positive dimensions, we get a map of spectra α 0 : S → H(Z) inducing an isomorphism on π 0 , where H(Z) is the Eilenberg-Maclane spectrum having all homotopy groups trivial except π 0 (H(Z)) ∼ = Z. The map α 0 is actually the 0-th Postnikov section of S (see Section 7). The stable Hurewicz homomorphism is the homomorphismh Definition 6.2. The long stable Whitehead exact sequence of a spectrum X is the homotopy exact sequence of the above cofibration: Here i is any integer,χ i is induced by (id ∧γ 0 ),h i is the stable Hurewicz homomorphism andν i is the connecting homomorphism. The groups π i (X ∧ S(0)) are usually denoted by Γ i (X): that definition coincides actually with the homotopy groups of the homotopy fiber of the Dold-Thom map (see [39]) and it was recently proved in [81, Corollary 3.9], that they are isomorphic to the groups 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 Γ i (X) by using the Atiyah-Hirzebruch spectral sequence for S(0)-homology (see [2,Section III.7]): This reproves the Hurewicz theorem because and consequentlyh i is an isomorphism for i ≤ b and an epimorphism for for any (b − 1)-connected spectrum X (this was already known by J. H. C. Whitehead for any (b − 1)-connected spectrum or for any (b − 1)-connected space with b ≥ 3, see [102, p. 81], or [101]). Thus, the homomorphismχ b+1 is actually a homomorphism from π b (X) ⊗ π 1 (S) to π b+1 (X). Consider the commutative diagram in which ∧ is the external product (see Definition 4.18 or [94, p. 270]). The top horizontal homomorphism is an isomorphism by Künneth formula and the two top vertical arrows, which are Hurewicz homomorphisms, 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 isomorphism. The homomorphism (id) * ⊗(γ 0 ) * is an isomorphism because (γ 0 ) * : π 1 (S(0)) Therefore,χ b+1 may be identified with the external product π b (X) ⊗ π 1 (S) ∧ −→ π b+1 (X). Thus, we proved the following result.
Our first goal is to show that the spectral sequence provides a generalization of that result for the exponent of all Gamma groups of X.

Now, if you look at the
Hirzebruch spectral sequence for a (b − 1)-connected spectrum X, it is obvious that the product of the exponents of the groups E 2 s,t , for s+t = i with t ≥ 1 and s ≥ b, kills the Gamma group Γ i (X). Because e t E 2 s,t = 0 for any t ≥ 1 by Definition 6.5, since π t (S(0)) ∼ = π t (S) when t ≥ 1, we conclude that the exponent of Γ i (X) divides the product e 1 e 2 e 3 · · · 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 for all integers i ≥ b + 1 and the stable Hurewicz homomorphismh i : Of course, we want to apply this theorem to the K-theory spectrum. Let us consider again the 0-connected K-theory spectrum X R of any ring R (see Definition 4.14) and let us kill its first homotopy group: we get the 1-connected K-theory spectrum X R (1). The above argument enables us to study the stable Hurewicz homomorphismh i : Theorem 6.6 holds here with b = 2. Corollary 6.7. For any ring R, the stable Hurewicz homomorphismh i : In particular, the exponent of the kernel, respectively of the cokernel, ofh i 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 integer i, the i-th algebraic K-group of R with coefficients in A is the i-th homotopy group of BGL(R) + or X R with coefficients in A (see [36] and [69]): In particular, if Z (p) denotes the ring of integers localized at p, then for all prime numbers p ≥ i 2 + 1. On the other hand, we can also deduce from the above considerations some information on the unstable Hurewicz homomorphism for i ≥ 2. Since BE(R) + is the 0-th space of the Ω-spectrum X R (1), we can look at the following commutative diagram for all integers i ≥ 2: where σ i is the iterated homology suspension (see [ If one works with coefficients in Z (p) , where p is a prime ≥ i 2 + 1, the composition σ i h i 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 Hurewicz homomorphism h i : K i (R; Z (p) ) → H i (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 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 homomorphism is contained in the kernel of the unstable Hurewicz homomorphism Proof: Let us denote by K Z (−1) the (−1)-connected K-theory spectrum of Z, the 0-th space of which is BGL(Z) + × K 0 (Z): it is a ring spectrum with unit η : S → K Z (−1) whose 0-connected cover S(0) → X Z is the map of spectra induced by the map of infinite loop spaces (BΣ ∞ ) + → BGL(Z) + which comes from the obvious inclusion of the infinite symmetric group Σ ∞ into GL(Z). This map η induces an isomorphism η * : π 1 (S) ∼ = −→ π 1 (K Z (−1)) ∼ = K 1 (Z) and the image of η * : π j (S) → K j (Z) for j ≥ 2 is described in [67] and [75]. Let R be any ring and for i ≥ 2, let us write X R (i − 1) for the (i − 1)-connected cover of the 0-connected K-theory spectrum X R . It is obvious that π j (X R (i−1)) ∼ = K j (R) for j ≥ i. By Definition 6.2 and Proposition 6.4, there is an exact sequence which commutes since K R (−1) is a K Z (−1)-module, shows that the above exact sequence is actually Now, let us write BGL(R) + (i − 1) for the (i − 1)-connected cover of the infinite loop space BGL(R) + and consider the commutative diagram where the iterated homology suspension σ i+1 is an isomorphism since i ≥ 2 (see [100,Corollary VII.6.5]). Thus, the composition is trivial and the assertion immediately follows if one composes h i+1 with the homomorphism induced by the obvious map 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 is trivial on all elements of the form x⊗y with x ∈ K i (R) and y belonging to the image of η * : π j (S) → K j (Z). In low dimensions, we are able to be more precise by providing exactness results. 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
Proof: (See [14,Theorem 4.1].) Let us consider the 1-connected infinite loop space BE(R) + and kill all its homotopy groups above dimension 3. We get its third Postnikov section (see also Section 7) BE(R) + [3] which has only two non-trivial homotopy groups π 2 (BE(R) + [3]) ∼ = K 2 (R) and π 3 (BE(R) + [3]) ∼ = K 3 (R). Therefore, BE(R) + [3] fits into the fibration of spaces in which the base space and the fiber are Eilenberg-MacLane spaces. Similarly, look at the third Postnikov section X R (1) [3] of the 1-connected cover X R (1) of X R and at the cofibration of spectra in which the base and the fiber are Eilenberg-MacLane spectra.
This induces the following commutative diagram where both columns are homology exact sequences: Here ∂ and∂ are connecting homomorphisms and the three horizontal arrows are iterated homology suspensions. Because of the long stable Whitehead exact sequence (see Definition 6.2 and Proposition 6.4), it turns out easily that and it is again possible to check that∂ is the product : [14]). Since σ 4 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 K 2 (R) K 1 (Z).
A similar argument provides the next theorem on the unstable Hurewicz homomorphism relating the algebraic K-theory of ring R to the homology of its infinite Steinberg group in dimensions 4 and 5.

Theorem 6.16. For any ring R there is a natural exact sequence
and the kernel of h 5 fits into the natural exact sequence where Q(R) is a quotient of the subgroup of elements of order dividing 2 in K 3 (R).
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 subject of the next section.

The Postnikov invariants in algebraic K-theory
The Postnikov invariants of a connected simple CW-complex X are cohomology classes which provide the necessary information for the reconstruction of X, up to a weak homotopy equivalence, from its homotopy groups. Let α i : X → X[i] denote the i-th Postnikov section of X for any positive 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 π j (X[i]) = 0 for j > i and (α i ) * : π j (X) → π j (X[i]) is an isomorphism for j ≤ i. Thus, we may view X[i] as the i-th homotopical approximation of X. The Postnikov k-invariants of X are cohomology classes for i ≥ 2, which are defined as follows (see for instance [100, Section IX.2]). Definition 7.1. Let X be a simple CW-complex, i an integer ≥ 2, and let κ i+1 denote the composition whereh i+1 is 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 given by the universal coefficient theorem and the homomorphism The main property of these invariants is that X[i] is the homotopy fiber of the map X[i − 1] → K(π i (X), i + 1) corresponding to the cohomology class k i+1 (X) ∈ H i+1 (X[i−1]; π i (X)), for i ≥ 2. In other words, there is a commutative diagram of fibrations in which the right column is the path fibration over the Eilenberg-MacLane space K(π i (X), i + 1) and the bottom square is a homotopy pull-back. Consequently, the knowledge of X[i − 1], π i (X) and k i+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 type of the K-theory space BGL(R) + of a ring R depends on the knowledge of the K-groups K i (R) = π i (BGL(R) + ) and of the k-invariants k i+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 k i+1 (BGL(R) + ) considered as elements of the group H i+1 (BGL(R) In order to understand the role of the k-invariants of a simple CW-complex, let us first mention the following obvious fact.
and the Hurewicz homomorphism h i : π i (X) → H i (X; Z) is split injective.
Proof: Since the diagram occuring in Definition 7.1 is a pull-back, the vanishing of k i+1 (X) implies that By definition of α i : X → X[i], the induced homomorphism (α i ) * : H j (X; Z) → H j (X[i]; Z) is an isomorphism for j ≤ i by the Whitehead theorem (see [100,Theorem IV.7.13]). Thus, the Künneth formula gives 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]).
Our first result is a vanishing theorem (see Theorem 7.6 below) based on the following remark on the cohomology suspension for Eilenberg-MacLane spaces.
Therefore, the assertion is a direct consequence of Proposition 7.5. See [8] for another proof.

Corollary 7.7. For any connected double loop space X,
where Λ 2 denotes the exterior square.
A direct application of that result to the infinite loop space BGL(R) + (see Remark 4.12) provides the following splitting (see also [11,Section 3], for the discussion of the naturality of that splitting).
This kind of nice consequences can be generalized when the k-invariant k i+1 (X) is a cohomology class which is not trivial, but of finite order in the group H i+1 (X[i − 1]; π i (X)). Proposition 7.10. Let X be a connected simple CW-complex, i an integer ≥ 2 and ρ a positive integer. The following assertions are equivalent: There is a homomorphism θ i : H i (X; Z) → π i (X) such that the composition is multiplication by ρ.
Therefore, we have the following commutative diagram where all columns are fibrations and in which the bottom left square is a pull-back by definition of the k-invariant k i+1 (X). Let E be the pull-back of (ρ k i+1 (X), p). Since the bottom composition in the above diagram is Since E is a pull-back, there is a map ϕ : X[i] → E inducing an isomorphism on π j for j ≤ i − 1 and multiplication by ρ on π i . Thus, we can define where the last map is the projection onto the second factor. This map induces multiplication by ρ on the only interesting homotopy group π i : Assertion (c) follows from (b) because of the commutativity of the diagram induced by the map f i , where both vertical arrows are Hurewicz homomorphisms: we call θ i the bottom horizontal homomorphism (f i ) * in that diagram.
In order to prove that (a) follows from (c), let us look at the commutative diagram in which the horizontal arrows are Hurewicz homomorphisms and the vertical arrows are connecting homomorphisms. If θ i : H i (X; Z) → π i (X) exists as in (c), we deduce that where κ i+1 is the element introduced in Definition 7.1. Thus, the image of ρκ i+1 under the isomorphism belongs to the image of the connecting homomorphism The exactness of the cohomology sequence Because of these equivalences, it is really important to prove finiteness results 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 calculation (see [38,Théorème 2]), the stable homotopy groups of Eilenberg-MacLane spaces have a quite small exponent. This can be formulated as follows.
This implies the following consequence.
such that there exists an integer t ≥ b with the property that π i (X) = 0 for i > t (in other words, such that X = X[t]). Then Proof: Let i be an integer such that t < i < 2b. If t = b, then X is an Eilenberg-MacLane space X = K(π b (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 whose Serre spectral sequence provides the exact sequence ; Z)= 0 and get the assertion because X[t] = X by hypothesis.
Definition 7.14. Let R j := 1 for j ≤ 1 and R j := j k=2 L k for j ≥ 2. For example, R 2 = 2, R 3 = 4, R 4 = 24, R 5 = 144, R 6 = 288 R 7 = 576, R 8 = 17280, . . . . It turns out that a prime number p divides R j 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 next result holds without any finiteness condition on the space we are looking at.
for all integers i such that 2 ≤ i ≤ r + 2b − 2.
Proof: Since X is (b − 1)-connected, it is clear that k i+1 (X) = 0 for 2 ≤ i ≤ b. Thus, we may assume that b + 1 ≤ i ≤ r + 2b − 2, in particular that r + b ≥ 3. It follows from the homotopy equivalence X Ω r Y that π i (X) ∼ = π i+r (Y ) and from Lemma 7.4 that the iterated cohomology suspension Since we may assume that Y is (b + r − 1)-connected, we deduce from Corollary 7.13 that for i + r − 1 < j < 2b + 2r, in particular for j = i + r and j = i + r + 1: Therefore, the universal coefficient theorem shows that the exponent of for all integers i ≥ 2.
In the case of the K-theory spaces, we get the following result. 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.
Let us look at immediate consequences of this theorem for the Hurewicz homomorphism 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 that this homomorphism is an isomorphism in the first non-trivial dimension (see Theorem 6.1). Our next result approximates the exponent of the kernel of the Hurewicz homomorphism in all dimensions (see also [9]). Remark 7.18. In [87, Proposition 3], C. Soulé has shown that the kernel of h i : K i (R) → H i (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 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 homomor- is multiplication by R i . If x belongs to the kernel of h i , then R i x = θ i h i (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 ones since it can be improved: for instance, in the case where i = 2, we know from Theorem 7.8 that h 2 : K 2 (R) → H 2 (GL(R); Z) is split injective for any ring R. H. Sah has also established that 2 ker h 3 = 0 for any ring A (see [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 is a split injection for all primes p ≥ i+3 2 . (b) For any integer i ≥ 2, the Hurewicz homomorphism is a split injection for all primes p ≥ i+2 2 . (c) For any integer i ≥ 3, the Hurewicz homomorphism is a split injection for all primes p ≥ i+1 2 . Proof: Let us look again at the composition is an isomorphism and h i is a split injection when p ≥ i+3 2 . The proof is analogous for the simply connected infinite loop space BE(R) + (with p dividing R i−1 if and only if p ≤ i+1 2 ) and for the 2-connected infinite loop space BSt(R) + (with p dividing R i−2 if 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 algebraically closed fields (see also [9,Theorem 2.4]).
Theorem 7.23. Let F be an algebraically closed field and i any positive even integer. Then, Proof: Since BSL(F ) + is a simply connected infinite loop space, we may consider an (i − 1)-connected space Y with BSL(F ) + Ω i−2 Y . By Lemma 7.4, the k-invariant k i+1 (BSL(F ) + ) is then the image of k 2i−1 (Y ) under the (i − 2)-fold iterated cohomology suspension Now, look at the universal coefficient theorem and observe that the group Ext(H 2i−2 (Y [2i − 3]; Z), K i (F )) vanishes because A. A. Suslin proved in Section 2 of [92] that K i (F ) is divisible for algebraically closed fields. Moreover, he also obtained in [92,Section 2], that K i (F ) is torsion-free if i is an even integer: this and the fact that

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 class 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 number field (i.e., a finite extension of the field of rationals Q) and O F its ring of algebraic integers. D. Quillen obtained in 1973 the first result on the structure of the groups K i (O F ) (see [73,Theorem 1]). Theorem 8.1 (Quillen). For any number field F and for any integer i ≥ 0, K i (O F ) is a finitely generated abelian group.
The corresponding result does not hold for the number field F itself: the structure of the abelian groups K i (F ) is much more complicated, and consequently much more interesting. Of course, the groups K i (F ) and K i (O F ) 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
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 O F and where the last map is induced by the inclusion O F → F . Here, for any ring R, P(R) is the category of finitely generated projective R-modules and BQP(−) denotes the Q-construction mentioned in Remark 3.16: in particular, its loop space fulfills the homotopy equivalence ΩBQP(R) BGL(R) + × K 0 (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 The homotopy exact sequence of that fibration provides the following long exact sequence.
Since O F /m is a finite field, the vanishing of K j (O F /m) whenever j is even ≥ 2 (see Corollary 5.9) then implies the following result.
(b) For any even integer i ≥ 2, there is a short exact sequence The next important information on the structure of the K-groups of number fields and rings of integers was obtained by A. Borel as a consequence of his study of the real cohomology of linear groups (see [31] or [32,Section 11]). Theorem 8.6 (Borel). Let F be a number field and let us write [F : Q] = r 1 + 2r 2 , where r 1 is the number of distinct embeddings of F into R and r 2 the number of distinct conjugate 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 O F , then the rational cohomology of the special linear group SL(R) is given by
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 A j and B k are the following exterior algebras: A j = Λ Q (x 5 , x 9 , x 13 , . . . , x 4l+1 , . . . ) and (b) If R denotes either the number field F or its ring of algebraic integers O F , then for any integer i ≥ 2, As a consequence, we observe: If R denotes a number field F or its ring of integers O F , then K i (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 the following statement.
However, the structure of the groups K i (F ) is quite complicated. In order to illustrate this, let us consider the subgroup D i (F ) of K i (F ) consisisting of all (infinitely) divisible elements in K i (F ) (notice that D i (F ) is not necessarily a divisible subgroup of K i (F )) and prove the following surprising assertion. Proof: Since K i (F ) is a finitely generated abelian group when i is odd according to Corollary 8.5, it does not contain any non-trivial divisible element and D i (F ) = 0. When i is even, consider again the localization exact sequence By Corollary 5.9, K i−1 (O F /m) is a finite cyclic group and contains therefore no non-trivial divisible elements. Consequently, the same is true for the direct sum m K i−1 (O F /m). It then follows that all divisible elements in K i (F ) actually belong to the image of the homomor- where ζ F (−) is the Dedekind zeta function of F , w m (k) the biggest integer s such that the exponent of the Galois group Gal(k(ξ s )/k) divides m for any field k (here ξ s is an s-th primitive root of unity), and F v 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 D 2m (Q) p .
Notice that the knowledge of D 2m (F ) is of particular interest since it is related to the Lichtenbaum-Quillen conjecture in algebraic K-theory (see Remark 9.16 and [24, Section II.2]) and toétale 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(O F ) + 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. The situation is more complicated for the special linear group SL(F ) over the number field F itself: in fact, the structure of the groups H i (SL(F ); Z) turns out to be similar to the structure of K i (F ) described in Corollary 8.8 (b). Theorem 8.12. For any number field F and any integer i ≥ 0, the group H i (SL(F ); Z) is the direct sum of a torsion group and a free abelian group of finite rank (which can be calculated by Theorem 8.6 (a)).
Proof: (See also [5,Section 2].) The proof is based on the results on the Postnikov invariants described in Section 7. Let C denote the Serre class of all abelian torsion groups. Because BSL(F ) + is a simply 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 which induces multiplication by the (finite) order of the corresponding k-invariant k i+1 (BSL(F ) + ) on each homotopy group π i (BSL(F ) In particular, f induces a C-isomorphism on each homotopy group. Now, let us compose f with the natural map which induces the quotient map (and thus a C-isomorphism) on each homotopy group. If we denote by Y this later space, this composition is a map inducing a C-isomorphism on all homotopy groups and therefore also a C-isomorphism on all integral homology groups because of the mod C Whitehead theorem (see [82,Section III.4]). On the other hand, since π i (Y ) ∼ = K i (F )/torsion is finitely generated for all integers i ≥ 1 by Corollary 8.8 (b), the homology groups H i (Y ; Z) of Y are also finitely generated. Consequently, the group H i (BSL(F ) + ; Z)/ ker ψ * ∼ = image ψ * is also finitely generated and ker ψ * belongs to C. If T i is written for the torsion subgroup of H i (BSL(F ) + ; Z), it follows that H i (BSL(F ) + ; Z)/T i is finitely generated, i.e., free abelian of finite rank, since it is a quotient of H i (BSL(F ) + ; Z)/ ker ψ * . Finally, this implies the vanishing of Ext(H i (BSL(F ) + ; Z)/T i , T i ) and the splitting of the extension The assertion then follows from the isomorphism Because of Theorem 8.3 (b), the groups K i (F ) are in general not finitely generated. By Serre class theory (see [82, Chapitre I]), this implies that the homology groups H i (SL(F ); Z) are in general not finitely generated. However, this only happens because of their torsion subgroups. The next step would be to investigate the structure of the torsion subgroups of the groups H i (SL(F ); Z). In particular, let us look at the subgroup D i (F ) of divisible elements in H i (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. Remark 8.14. Observe that for any number field F , the homomorphism  [18]). However, one can prove (see Theorem 1.4 of [18]) the injectivity of the induced homomorphism H i (SL(O F ); Z (p) ) → H i (SL(F ); Z (p) ) in small dimensions, more precisely for 2 ≤ i ≤ min(2p − 2, d p (F ) + 1), where d p (F ) denotes the smallest positive integer j for which K j (F ) contains non-trivial p-torsion divisible elements (d p (F ) is an even integer according to Theorem 8.9 and we say that d p (F ) = ∞ if there are no p-torsion divisible elements in K j (F ) for all j ≥ 1).

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 ring of integers Z, we first know that E(Z) = SL(Z) by Theorem 2.10 and we may deduce from Section 8 the following result on the structure of the abelian groups K i (Z). Proof: Theorem 8.1 asserts that the groups K i (Z) are finitely generated abelian groups for all i ≥ 0. Moreover, the rank of the free abelian group K i (Z)/torsion is given by Theorem 8.6 (b), with r 1 = 1 and r 2 = 0.
In low dimensions, the K-groups of Z have been computed for i ≤ 4 and partially determined for i = 5.
Proof: See Theorem 1.2, Theorem 2.10 and Example 2.29 for the calculation of K 0 (Z), K 1 (Z) and K 2 (Z), [55] for the determination of K 3 (Z), [76], [88], [77], [99] and [95] for the vanishing of K 4 (Z), and [56] and [84] for the description of K 5 (Z). Let us also look at the unstable Hurewicz homomorphisms Theorem 9.4. The following sequences are exact: Proof: The unstable Whitehead exact sequence (see [102]) of the simply connected space BSL(Z) + is 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 K i (Z) → K i (F p ) induced by the reduction mod p : Z → F p for various primes p, he could prove the following relationship between the order of the torsion subgroups of the K-groups of Z and the denominators of the Bernoulli numbers.
Definition 9.7. The Bernoulli numbers are the rational numbers B m occuring in the power series . It is not hard to check that The first Bernoulli numbers are 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 K 4j−1 (Z) contains a cyclic subgroup Q 4j−1 of order 2E 2j . If j is even, Q 4j−1 is a direct summand of K 4j−1 (Z). If j is odd, the odd-torsion part of Q 4j−1 is a direct summand of K 4j−1 (Z) and the 2-torsion part of Q 4j−1 (which is ∼ = Z/8) is contained in a cyclic direct summand of order 16 of K 4j−1 (Z).
Proof: See [75] for the detection of the subgroup Q 4j−1 of order 2E 2j in K 4j−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. Another very surprising result was proved by C. Soulé in 1979 when he explained that in fact the numerators of the Bernoulli numbers also play a role in the investigation of the torsion in the groups K i (Z). Recall the following definition. . It turns out that p is irregular if and only if p divides the class number of the cyclotomic field Q(ξ p ), where ξ p is a p-th primitive root of unity; moreover, an irregular prime p is called properly irregular if p does not divide the class number of the maximal real subfield of Q(ξ 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 (Soulé). If p is a properly irregular prime number and if m is a positive even integer < p such that p divides the numerator of B m+1 m + 1 , then the algebraic K-theory group K 2m (Z) contains an element whose order is equal to the p-primary part of B m+1 m + 1 .
Proof: See [85, Section IV.3, Théorème 6], where the argument is based on the investigation of the relationships between algebraic K-theory and etale cohomology.
Remark 9.15. Consider again the short exact sequences given by Theorem 8.3 for all positive integers m. The torsion elements of K 2m (Z) detected by C. Soulé and presented in Theorem 9.13 also belong to the group K 2m (Q) and play a special role in that group with respect to the subgroup D 2m (Q) of divisible elements in K 2m (Q) (see Theorem 8. Thus, apart from some classes of order 2, the known torsion classes in the algebraic K-theory of Z which occur in odd degrees, respectively in even degrees, are related to the denominators, respectively to the numerators, of the Bernoulli numbers. The next attempt to understand the K-theory of Z was made by M. Bökstedt in 1984 (see [30]): he tried to construct a model for the algebraic K-theory space BGL(Z) + and proved that this model detects the known torsion classes at the prime 2. His idea was simple and excellent: he considered 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(F p ) + . Then, he introduced a space J(p) which is However, K i−1 (F 2 ) ⊗ Z ∧ 2 is always trivial according to Corollary 5.9 except if i = 1, where K 0 (F 2 )⊗Z ∧ 2 ∼ = Z ∧ 2 . Therefore, (BGL(Z) + ×S 1 ) ∧ 2 ∼ = (BGL(Z[ 1 2 ]) + ) ∧ 2 and ψ is actually a map (BGL(Z[ 1 2 ]) + ) ∧ 2 → J(p) ∧ 2 which induces also a split surjection on all homotopy groups.
A very significant step was made by V. Voevodsky in 1997, when he proved the Milnor conjecture [95] which asserts that if F is a field of characteristic = 2, then K M i (F )/2K M i (F ) ∼ = H í et (F ; Z/2) (see Definition 2.33). This fundamental theorem has many deep consequences. In particular, J. Rognes and C. Weibel were then able to use it in order to calculate the E 2 -term of the Bloch-Lichtenbaum spectral sequence E s,t 2 =⇒ K −s−t (Q; Z/2) and, after a very tricky study of its differentials, to determine the groups K i (Q; Z/2). They could then deduce from the localization exact sequence the calculation of K i (Z; Z/2) for all integers i. At that point, they were very lucky since all elements of the groups K i (Z; Z/2) were detected by the elements of K i (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-torsion of the groups K i (Z). Theorem 9.18 (Voevodsky, Rognes-Weibel). K 1 (Z) ∼ = Z/2 and for i ≥ 2, This theorem has an immediate crucial topological consequence. Consider any prime p ≡ 3 or 5 mod 8 and the above map ψ : (BGL(Z) + ) ∧ 2 → JK(Z, p) ∧ 2 which induces a split surjection on all homotopy groups. It is easy to check that Theorem 9.18 shows that the homotopy groups of (BGL(Z) + ) ∧ 2 and of JK(Z, p) ∧ 2 are the same. Therefore, Theorem 9.17 implies that the induced homomorphism ψ * : π i ((BGL(Z) + ) ∧ 2 ) → π i (JK(Z, p) ∧ 2 ) is an isomorphism for all positive integers i. This and a similar argument for the map Corollary 9.25. For all integers i ≥ 2, the 2-primary part (ρ i ) 2 of the integer ρ i is given by Proof: This follows directly from Proposition 7.10 and the description of the Hurewicz homomorphism at the prime 2 provided by Corollary 9.22 and Remark 9.23 (see [21,Théorème 5.15] for more details).
The fibration given by Theorem 9.21 enables us to deduce two other important properties of the 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 where the first arrow is the usual K-theoretical product defined in Definition 4.5 and the second the tensor product of K i+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 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 two elements of order 2.
for i ≡ 2 mod 8, which is useful in order to calculate the 2-adic product in the three remaining cases of the above list (see [14,Theorems 5.6,5.7,5.8 and 5.9] for the details of all these computations).
Finally, the fibration also provides the determination of the Hopf algebra structure of the mod 2 cohomology of the infinite general linear group GL(Z) as a module over the Steenrod algebra. Proof: The recent paper [15] contains the proof of this assertion (see Theorem 1 of [15]) and an explicit definition of the exterior classes u 2k−1 ∈ H 2k−1 (BGL(Z) + ; Z/2) (see [15,Definition 10 and Remark 14]). The classes w j are the images under θ * : H * (BO; Z/2) → H * (BGL(Z) + ; Z/2) of the universal Stiefel-Whitney classes in H * (BO; Z/2) (see [66,Chapter 7]). Notice that an additive version of the above isomorphism has been conjectured in [42,Corollary 4.3], and that the statement of the theorem 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-theory 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 Bökstedt'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/l 2 ) * , then the l-completion JK(Z, p) ∧ of JK(Z, p) does not depend on p (see [21,Proposition 3.24]): we shall denote it by JKZ ∧ . Corollary 9.34. If l is a regular prime and if the Lichtenbaum-Quillen conjecture is true at l (see Remark 9.31), then the integers (ρ i ) l can be exactly determined: is a proper divisor of (i + 1), 1, otherwise, except for (ρ 11 ) 3 which is equal to 3.

Further developments
The goal of this paper was to show how some topological methods can provide very general and deep results on the algebraic K-theory of rings. We especially emphasized the use of the infinite 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 homotopical approximations. Of course, the arguments presented here do not represent all the topological considerations which can give interesting K-theoretical information.
It is not our purpose to describe these other ideas in details in this paper, but 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 F p for any prime p and to deduce nice theorems on the ordinary algebraic K-theory: in particular, he could exhibit a periodicity result for the groups K i (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, theétale K-theory spectrum of R, whose homotopy groups are called theétale K-groups of R: they are in principle easier to calculate and some strong results are known on their relationships with the ordinary algebraic K-groups of R; however, the homomorphism relating these two K-theories is still the object of several difficult conjectures (see [41] and [42] for example). Some other impressive progress has been made by using techniques from stable homotopy theory (see for instance the works by M. Bökstedt, W. G. Dwyer, R. McCarthy, S. A. Mitchell, J. Rognes and C. Weibel).
Let us finally recall that the definition of the algebraic K-theory of a ring R is based on the Q-construction over the category P(R) of finitely generated projective R-modules (see Remark 3.16): of course, this can also be done if we replace the category of finitely generated projective modules over a ring by another nice category. This shows that the main specificity of algebraic K-theory comes from its interaction with other areas of mathematics. Therefore, many K-theoretical results are consequences of methods from algebraic geometry, arithmetic, number theory, algebra, cyclic homology, operator theory, and so on. This explains also why almost all problems in algebraic K-theory are quite complicated and why many conjectures are still unsolved. However, this is obviously a sign of the great role that algebraic K-theory will play in the future.