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34
Algebraic Ktheory of topological Ktheory
"... Let ℓp be the pcomplete connective Adams summand of topological Ktheory, 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 Ktheory spectrum of ℓp, denoted V (1)∗K(ℓp ..."
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Cited by 40 (16 self)
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Let ℓp be the pcomplete connective Adams summand of topological Ktheory, 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 Ktheory 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 Ktheory increases chromatic complexity by one. The proof uses the cyclotomic trace map from algebraic Ktheory to topological cyclic homology, and the calculation is
Quasisymmetric functions from a topological point of view arXiv: math.AT/0605743
"... Abstract. It is wellknown 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 topolo ..."
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Cited by 24 (2 self)
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Abstract. It is wellknown 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 noncommutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.
Topological Hochschild homology of Thom spectra which are . . .
, 2008
"... 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 Salgebra ..."
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Cited by 23 (9 self)
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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 Salgebra (E ∞ ring spectrum) R can be described as an indexed colimit together with a verification that the LewisMay 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).
Topological Hochschild homology of connective complex Ktheory
 Amer. J. Math
"... Abstract. Let ku be the connective complex Ktheory spectrum, completed at an odd prime p. We present a computation of the mod (p, v1) homotopy algebra of the topological Hochschild homology spectrum of ku. 1. Introduction. Since ..."
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Cited by 16 (1 self)
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Abstract. Let ku be the connective complex Ktheory spectrum, completed at an odd prime p. We present a computation of the mod (p, v1) homotopy algebra of the topological Hochschild homology spectrum of ku. 1. Introduction. Since
Hopf algebra structure on topological Hochschild homology
, 2005
"... The topological Hochschild homology THH(R) of a commutative Salgebra (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) in ..."
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Cited by 15 (8 self)
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The topological Hochschild homology THH(R) of a commutative Salgebra (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 Salgebras 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 Ktheory of Salgebras, using topological cyclic homology.
Differentials in the homological homotopy fixed point spectral sequence
, 2005
"... Abstract We analyze in homological terms the homotopy fixed point spectrum of a Tequivariant commutative Salgebra 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 homot ..."
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Cited by 11 (3 self)
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Abstract We analyze in homological terms the homotopy fixed point spectrum of a Tequivariant commutative Salgebra 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 2rterm of the spectral sequence there are 2r other classes in the E 2rterm (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 Salgebras, 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 Ktheory of commutative Salgebras.
On the algebraic Ktheory of the complex Ktheory spectrum
, 2006
"... Abstract. Let p � 5 be a prime, let ku be the connective complex Ktheory spectrum, and let K(ku) be the algebraic Ktheory spectrum of ku. In this paper we study the pprimary homotopy type of the spectrum K(ku) by computing its mod (p, v1) homotopy groups. We show that up to a finite summand, thes ..."
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Cited by 11 (1 self)
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Abstract. Let p � 5 be a prime, let ku be the connective complex Ktheory spectrum, and let K(ku) be the algebraic Ktheory spectrum of ku. In this paper we study the pprimary 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.
Topological Hochschild cohomology and generalized Morita equivalence, Algebraic & Geometric Topology 4
, 2004
"... 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 ..."
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Cited by 10 (1 self)
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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 Salgebra R is an EilenbergMacLane spectrum of a commutative ring this notion specializes to classical Azumaya algebras. A canonical example of a topological Azumaya Ralgebra is the endomorphism Ralgebra FR(M, M) of a finite cell Rmodule. We show that the spectrum of mod 2 topological Ktheory KU/2 is a nontrivial topological Azumaya algebra over the 2adic completion of the Ktheory 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 Salgebra A.
Γcohomology of rings of numerical polynomials and E∞ structures on
"... Abstract. We investigate Γcohomology 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 Γcohomology vanishes above degree 1. As these cohomology groups are the obstruction group ..."
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Cited by 8 (7 self)
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Abstract. We investigate Γcohomology 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 Γcohomology 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 JohnsonWilson spectrum E(n) with n � 1 we establish the existence of a unique E ∞ structure for its Inadic completion.