The celebrated Lichtenbaum-Quillen conjectures predict that for a sufficiently nice scheme and given prime ℓ, the ℓ-adic algebraic K-groups of X are closely related to the ℓ-adic étale cohomology groups of X. More precisely, one version of the conjectures asserts that there is a descent spectral sequence of Atiyah-Hirzebruch type H p q ét (X; Zℓ ()) ⇒ πq−p(KX 2 ℓ) but with the convergence only valid in sufficiently high degrees. Here the coefficient sheaves are Tate twists of the ℓ-adic integers, and are to be interpreted as zero if q is odd. Throughout this paper, étale cohomology is continuous étale cohomology [19], and the indicated abutment of the spectral sequence consists of the homotopy groups of the Bousfield ℓ-adic completion of the spectrum KX, not the naive ℓ-adic completion of the K-groups. In a remarkable paper [42], Thomason proved the Lichtenbaum-Quillen conjectures for a certain localized form of K-theory- so-called “Bott-periodic” K-theory. The first step was the development of an elaborate theory of hypercohomology spectra H ·(X; E) associated to étale presheaves of spectra E- or more generally, to presheaves of spectra on a Grothendieck site. These hypercohomology spectra are by their very construction naturally Supported by a grant from the National Science Foundation 1 equipped with a suitable descent spectral sequence, and there is a natural map E(X)−→H · (X; E). In particular, the K-theory of a scheme X maps to its associated hypercohomology spectrum H · ét(X; K). Fix a prime ℓ, which we will assume is odd in order to simplify the discussion.Let L(−) denote Bousfield localization with respect to complex K-theory, and let ˆ L(−) denote its ℓ-adic completion. The main theorem of [42] can be stated as follows. Theorem 0.1 Let X be a separated noetherian regular scheme of finite Krull dimension, with sufficiently nice residue fields of characteristic different from ℓ. Then the natural map KX−→H · ét(X; K) induces a weak equivalence

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Introduction One of the central problems of algebraic K-theory is to compute the K-groups K n X of a scheme X. Since these groups are, by denition, the homotopy groups of a spectrum KX, it makes sense to analyze the homotopy-type of the spectrum, rather than just the disembodied homotopy groups. In addition to facilitating the computation of the K-groups themselves, knowledge of the spectrum KX can be applied to the study of other topological invariants. For example, if X = Spec R, then the homology groups of the zero-th space 1 KX are of interest since they are the homology groups of the innite general linear group GLR; but they are not determined by the homotopy groups of KX alone. Topological complex K-theory is another important invariant. Let K denote the periodic complex K-theory spectrum, and let ^ K denote its Bouseld `-adic completion

2. Foundational results 6 3. Divisibility of motivic cohomology; proof of theorem 3 a) 8

Abstract. Let p be an odd regular prime, and assume that the Lichtenbaum– Quillen conjecture holds for K(Z[1/p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum W h(∗) is described. A suspended copy of the cokernel-of-J spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted S 1-transfer map t: ΣCP ∞ → S. The homotopy of W h(∗) is determined in a range of degrees, and the cohomology of W h(∗) is expressed as an A-module in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or diffeomorphisms of highly connected, high dimensional compact smooth manifolds.

We compute the mod 2 cohomology of Waldhausen’s algebraic K-theory spectrum A(∗) of the category of finite pointed spaces, as a module over the Steenrod algebra. This also computes the mod 2 cohomology of the smooth Whitehead spectrum of a point, denoted WhDiff (∗). Using an Adams spectral sequence we compute the 2-primary homotopy groups of these spectra in dimensions ∗ ≤ 18, and up to extensions in dimensions 19 ≤ ∗ ≤ 21. As applications we show that the linearization map L: A(∗) → K(Z) induces the zero homomorphism in mod 2 spectrum cohomology in positive dimensions, the space level Hatcher–Waldhausen map hw: G/O → ΩWh Diff (∗) does not admit a four-fold delooping, and there is a 2-complete spectrum map M: Wh Diff (∗) → Σg/o ⊕ which is precisely 9-connected. Here g/o ⊕ is a spectrum whose underlying space has the 2-complete homotopy type of G/O.

Abstract. We compute the two-completed algebraic K-groups K∗ ( ˆ Z2) ∧ 2 of the twoadic integers, and determine the homotopy type of the two-completed algebraic K-theory spectrum K ( ˆ Z2) ∧ 2. The natural map K(Z) ∧ 2 → K( ˆ Z2) ∧ 2 is shown to induce an isomorphism modulo torsion in degrees 4k + 1 with k ≥ 1.

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the prime 2

this paper; in effect we work entirely in the category H