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

We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E ∞ classifying map X → BG, for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative S-algebra (E ∞ ring spectrum) R can be described as an indexed colimit together with a verification that the Lewis-May operadic Thom spectrum functor preserves indexed colimits. We prove a splitting result THH(Mf) ≃ Mf ∧BX+ which yields a convenient description of THH(MU). This splitting holds even when the classifying map f: X → BG is only a homotopy commutative A ∞ map, provided that the induced multiplication on Mf extends to an E ∞ ring structure; this permits us to recover Bokstedt’s calculation of THH(HZ).

Abstract. We investigate Γ-cohomology of some commutative cooperation algebras E∗E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Γ-cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E ∞ structures. As a consequence we obtain an E∞ structure for the connective Adams summand. For the Johnson-Wilson spectrum E(n) with n � 1 we establish the existence of a unique E ∞ structure for its In-adic completion.

The topological Hochschild homology THH(R) of a commutative S-algebra (E ∞ ring spectrum) R naturally has the structure of a Hopf algebra over R, in the homotopy category. We show that under a flatness assumption this makes the Bökstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to study some interesting examples, including the commutative S-algebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic K-theory of S-algebras, using topological cyclic homology.

Abstract. It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions Symm. We offer the cohomology of the space ΩΣCP ∞ as a topological model for the ring of quasisymmetric functions QSymm. We exploit standard results from topology to shed light on some of the algebraic properties of QSymm. In particular, we reprove the Ditters conjecture. We investigate a product on ΩΣCP ∞ that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of BU. The canonical Thom spectrum over ΩΣCP ∞ is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.

Abstract. We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when M is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer & Greenlees and of Schwede & Shipley. A central notion of noncommutative ring theory related to Morita equivalence is that of central separable or Azumaya algebras. For such an Azumaya algebra A, its Hochschild cohomology HH ∗ (A,A) is concentrated in degree 0 and is equal to the center of A. We introduce a notion of topological Azumaya algebra and show that in the case when the ground S-algebra R is an Eilenberg-MacLane spectrum of a commutative ring this notion specializes to classical Azumaya algebras. A canonical example of a topological Azumaya R-algebra is the endomorphism R-algebra FR(M, M) of a finite cell R-module. We show that the spectrum of mod 2 topological K-theory KU/2 is a nontrivial topological Azumaya algebra over the 2-adic completion of the K-theory spectrum ̂ KU2. This leads to the determination of THH(KU/2, KU/2), the topological Hochschild cohomology of KU/2. As far as we know this is the first calculation of THH(A, A) for a noncommutative S-algebra A.

Abstract. Let p � 5 be a prime, let ku be the connective complex K-theory spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. In this paper we study the p-primary homotopy type of the spectrum K(ku) by computing its mod (p, v1) homotopy groups. We show that up to a finite summand, these groups form a finitely generated free module over a polynomial algebra Fp[b], where b is a class of degree 2p + 2 defined as a “higher Bott element”. 1.

Abstract We analyze in homological terms the homotopy fixed point spectrum of a T-equivariant commutative S-algebra R. There is a homological homotopy fixed point spectral sequence with E2 s,t = H−s gp (T; Ht(R; Fp)), converging conditionally to the continuous homology H c s+t (RhT; Fp) of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations β ǫ Q i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the E 2r-term of the spectral sequence there are 2r other classes in the E 2r-term (obtained mostly by Dyer–Lashof operations on x) that are infinite cycles, i.e., survive to the E ∞-term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH(B) of many S-algebras, including B = MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups C ⊂ T, and for the Tate- and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic K-theory of commutative S-algebras.

Abstract. We analyze the homotopy fixed point spectrum of a T-equivariant commutative S-algebra R in homological terms. There is a homological homotopy fixed point spectral sequence with E 2 s,t = H−s (T; Ht(R; Fp)), which converges conditionally to the continuous homology H c ∗ (RhT; Fp) of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations β ǫ Q i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the E 2r-term of the spectral sequence there are 2r other classes in the E 2r-term (obtained mostly by Dyer–Lashof operations on x) that are infinite cycles, i.e., survive to the E ∞-term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH(B) of many S-algebras, including B = MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups C ⊂ T, and for the Tate- and homotopy orbit spectra. This work is part of a homological approach to calculating topological cyclic homology and algebraic K-theory of commutative S-algebras. 1.

Abstract. We calculate the integral homotopy groups of THH (ℓ) at any prime and of THH (ko) at p = 2, where ℓ is the Adams summand of the connective complex p-local K-theory spectrum and ko is the connective real K-theory spectrum. 1. Introduction. 1.1. Motivation. Topological Hochschild homology is a generalization of Hochschild homology to the context of structured ring spectra. In analogy with Hochschild homology, it helps classifying deformations and extensions of structured ring spectra.