Let ℓp be the p-complete connective Adams summand of topological K-theory, with coefficient ring (ℓp) ∗ = Zp[v1], and let V (1) be the Smith–Toda complex, with BP∗(V (1)) = BP∗/(p, v1). For p ≥ 5 we explicitly compute the V (1)-homotopy of the algebraic K-theory spectrum of ℓp, denoted V (1)∗K(ℓp). In particular we find that it is a free finitely generated module over the polynomial algebra P (v2), except for a sporadic class in degree 2p − 3. Thus also in this case algebraic K-theory increases chromatic complexity by one. The proof uses the cyclotomic trace map from algebraic K-theory to topological cyclic homology, and the calculation is

This paper calculates the relative algebraic K-theory K∗(k[x]/(xn), (x)) of a truncated polynomial algebra over a perfect field k of positive characteristic p. Since the ideal generated by x is nilpotent, we can apply McCarthy’s theorem: the relative algebraic K-theory is isomorphic to the relative topological cyclic homology, [Mc], and it is the latter groups we actually evaluate.

Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an Eilenberg-Mac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are topologically equivalent, but we produce explicit counterexamples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting. Contents

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.

Abstract. We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of Thomason-Trobaugh in K-theory. We also deduce versions of Thomason’s blow-up formula and the projective bundle formula for THH and TC. 1.

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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.

Let p be any prime. We consider Bökstedt’s topological refinement K(Z) → T (Z) = T HH(Z) of the Dennis trace map from algebraic K-theory of the integers to topological Hochschild homology of the integers. This trace map is shown to induce a surjection on homotopy in degree 2p − 1, onto the first p-torsion in the target. Furthermore, Bökstedt’s map factors through the S 1-homotopy fixed points T (Z) hS1 of T (Z), and it is shown that the first p-torsion element in degree 2p − 3 of the stable homotopy groups of spheres is detected in the homotopy of T (Z) hS1 Both results are due to Bökstedt, but have remained unpublished.

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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