HERMITIAN K-THEORY OF THE INTEGERS By A. J. BERRICK and M. KAROUBI

Rognes and Weibel used Voevodsky’s work on the Milnor conjecture to deduce the strong Dwyer-Friedlander form of the Lichtenbaum-Quillen conjecture at the prime 2. In consequence (the 2-completion of) the classifying space for algebraic K-theory of the integers Z[1/2] can be expressed as a fiber product of well-understood spaces BO and BGL(F3) over BU. Similar results are now obtained for Hermitian K-theory and the classifying spaces of the integral symplectic and orthogonal groups. For the integers Z[1/2], this leads to computations of the 2-primary Hermitian K-groups and affirmation of the Lichtenbaum-Quillen conjecture in the framework of Hermitian K-theory. 0. Introduction. In [9], Bökstedt introduced the study of the commuting square BGL(Z)# −→ BGL(R)# ↓ ↓ BGL(F3)# b −→ BGL(C)#. (0-1) Here Z′ denotes the ring Z[1/2], F3 the finite field with three elements, and, for F = R,C, BGL(F) is the classifying space of the infinite general linear group GL(F) with the usual topology. The symbol # indicates the 2-adic completion, and the map b the Brauer lift, corresponding to the fibring of Adams’ map ψ3−1 on BU = BGL(C). The remaining maps are induced from the obvious ring homomorphisms. The Dwyer-Friedlander formulation of the Lichtenbaum-Quillen conjecture for Z at the prime 2 is that the above square is homotopy cartesian [15] Conjecture 1.3, Proposition 4.2. This has been affirmed in work of Rognes and Weibel [48]— see [57] Corollary 8. Since the homotopy fiber of the map BGL(R) → BGL(C) is the homogeneous space GL(C)/GL(R), which has the homotopy type of Ω7(BGL(R)) by Bott periodicity [11], we may also write the homotopy fibration Ω(BGL(R))# −→ BGL(Z)# −→ BGL(F3)#. The purpose of this paper is to prove analogous results (see Section 2 below) with the orthogonal and symplectic groups over Z′ substituted for the general linear Manuscript received October 3, 2003; revised January 13, 2005. American Journal of Mathematics 127 (2005), 785–823. 785 786 A. J. BERRICK AND M. KAROUBI group. Our motivation comes from previous work by the first author involving the mapping class group [8], and by the second author on the analogue of Bott periodicity in Hermitian K-theory [26]. This subject is of course related to the computations by A. Borel [10] of the rational cohomology of arithmetic groups. Acknowledgments. We thank A. Bak, L. Fajstrup, E. M. Friedlander, H. Hamraoui, J. Hornbostel, B. Kahn and L. N. Vaserstein for their kind interest in this work. In this regard we would like to acknowledge the conscientiousness of L. Fajstrup, whose recent amendment to some calculations of [16] §8 provides independent confirmation of a key point in our proof of Theorem C of Section 2. 1. Motivational background. The commutative diagram below appears in [8]. In it, Brg denotes the g-strand braid group and the map Artin is Artin’s representation of Brg as automorphisms of the free group Frg on g generators. The group Γg,1 is the mapping class group of a surface of genus g with one boundary component. The map ψg is constructed by Vershinin [54] p. 1000. H is the hyperbolic map sending a matrix A to the matrix ( A O O tA−1 ) . Γg,1 id −→ Γg,1 ↗ ψg ↘ Brg Σg Og(Z) H −→ Sp2g(Z) Artin ↘ ↓ ↓ Aut(Frg) GLg(Z) H −→ GL2g(Z). Here all maps are injective except for the surjections Aut(Frg) GLg(Z), Brg Σg and Γg,1 Sp2g(Z). Now, as in [8], combine with the inclusions in the real symplectic and general linear groups, stabilize, and take B( · )+, to get BΓ∞ id → BΓ∞ BU ↗ ↘ ↓ BBr∞ → BΣ∞ → BO(Z)+ → BSp(Z)+ → BSpR ↘ ↓ ↓ ↓ BAut(Fr∞) → BGL(Z)+ → BGL(Z)+ → BGLR. The challenge is to describe these maps in homotopy theory. For instance, [14] discusses BΓ∞ → BGL(Z)+, while [36] looks at BΓ∞ → BU. In this work we focus on BO(Z′)+ and BSp(Z′)+ as accessible approximations to BO(Z)+ and BSp(Z)+. Indeed, if we extrapolate the results known in the higher Hermitian K-theory of rings A (where 2 is invertible) and the results known in lower K-theory, then it is reasonable to conjecture that, up to odd finite torsion, HERMITIAN K-THEORY OF THE INTEGERS 787 for i > 1 we have the isomorphisms πi(BO(Z) ) ∼= πi(BO(Z)) and πi(BSp(Z)) ∼= πi(BSp(Z)). 2. Main results. We recall first some notations from [26]. Let A be a ring provided with an anti-involution x → x̌, and let ε be an element of the center of A such that εε̌ = 1. We assume also the existence of an element λ of the center such that λ + λ̌ = 1. In most cases, λ = 1/2 and ε = ±1. In this setting, a central role is played by the ε-orthogonal group, which is the group of automorphisms of the ε-hyperbolic module εH(A), denoted by εOn,n(A): its elements can be described as 2 × 2 matrices written in n-blocks