Hermitian K -theory of the integers

Abstract

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 wellunderstood spaces BO and BGL(F 3 ) + 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 Kgroups and affirmation of the Lichtenbaum-Quillen conjecture in the framework of Hermitian K-theory.Here Z ′ denotes the ring Z[1/2], F 3 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 ψ 3 − 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 Ω 7 (BGL(R)) by Bott periodicity [11], we may also write the homotopy fibration Ω 7 (BGL(R)) # −→ BGL(Z ′ ) + # −→ BGL(F 3 ) + # .

Slices of hermitian K–theory and Milnor’s conjecture on quadratic forms

“…Computations similar to the above yields the E 2 -page: We read off that KSp 0 (F ) ∼ = 2H 0,0 is infinite cyclic and KSp 1 (F ) is the trivial group. It follows that all the classes in the sixth column are infinite cycles and E 2 p,q = E ∞ p,q when (p, q) = (6, 4), (6,5). Hence we obtain a surjection KSp 2 (F ) ≻ H 2,2 .…”

“…A closely related statement is obtained in [49]. The sequence (5) is employed in our computation of the slices of KQ.…”

“…To prove Mµ is trivial we use the solution of the homotopy limit problem for hermitian K-theory of prime fields due to Berrick-Karoubi [5] and Friedlander [11]. Their results have been generalized by Hu-Kriz-Ormsby [20] and Berrick-Karoubi-Schlichting-Østvaer [6], where the following is shown.…”

“…Our first results show there is an additional "mysterious summand" Σ 2q,q Mµ of s q (KQ). We show Mµ is trivial by using base change and the solution of the homotopy fixed point problem for hermitian K-theory of the prime fields [5], [11], cf. [6], [20].…”

Algebraic and Hermitian K-Theory of -Rings

“…This parallels the construction of the algebraic K-theory spectrum K(A) for a unital ring. (Note that we use notation KQ instead of L, the notation employed in [21] and [7], in order to avoid confusion with the surgery spectrum and the surgery groups. )…”