118 research outputs found
The circle action on topological Hochschild homology of complex cobordism and the Brown-Peterson spectrum
We specify exterior generators for and , and calculate the action of the -operator on these graded
rings. In particular, and ,
while the actions on and are expressed in terms of the
right units in the Hopf algebroids
and , respectively.Comment: This paper has been accepted for publication by the Journal of
Topolog
Stably dualizable groups
We extend the duality theory for topological groups from the classical theory
for compact Lie groups, via the topological study by J. R. Klein [Kl01] and the
p-complete study for p-compact groups by T. Bauer [Ba04], to a general duality
theory for stably dualizable groups in the E-local stable homotopy category,
for any spectrum E. The principal new examples occur in the K(n)-local
category, where the Eilenberg-Mac Lane spaces G = K(Z/p, q) are stably
dualizable and nontrivial for 0 <= q <= n.
We show how to associate to each E-locally stably dualizable group G a stably
defined representation sphere S^{adG}, called the dualizing spectrum, which is
dualizable and invertible in the E-local category. Each stably dualizable group
is Atiyah-Poincare self-dual in the E-local category, up to a shift by S^{adG}.
There are dimension-shifting norm- and transfer maps for spectra with G-action,
again with a shift given by S^{adG}. The stably dualizable group G also admits
a kind of framed bordism class [G] in pi_*(L_E S), in degree dim_E(G) =
[S^{adG}] of the Pic_E-graded homotopy groups of the E-localized sphere
spectrum.Comment: Final version, to appear in the Memoirs of the A.M.
Algebraic K-theory of strict ring spectra
We view strict ring spectra as generalized rings. The study of their
algebraic K-theory is motivated by its applications to the automorphism groups
of compact manifolds. Partial calculations of algebraic K-theory for the sphere
spectrum are available at regular primes, but we seek more conceptual answers
in terms of localization and descent properties. Calculations for ring spectra
related to topological K-theory suggest the existence of a motivic cohomology
theory for strictly commutative ring spectra, and we present evidence for
arithmetic duality in this theory. To tie motivic cohomology to Galois
cohomology we wish to spectrally realize ramified extensions, which is only
possible after mild forms of localization. One such mild localization is
provided by the theory of logarithmic ring spectra, and we outline recent
developments in this area.Comment: Contribution to the proceedings of the ICM 2014 in Seou
Hopf algebra structure on topological Hochschild homology
The topological Hochschild homology THH(R) of a commutative S-algebra
(E_infty ring spectrum) R naturally has the structure of a commutative
R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy
category. We show, under a flatness assumption, that this makes the Boekstedt
spectral sequence converging to the mod p homology of THH(R) into a Hopf
algebra spectral sequence. We then apply this additional structure to the study
of 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, by means of
the cyclotomic trace map to topological cyclic homology.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-49.abs.htm
Algebraic K-theory of the first Morava K-theory
We compute the algebraic K-theory modulo p and v_1 of the S-algebra ell/p =
k(1), using topological cyclic homology.Comment: Revised version, to appear in J. Eur. Math. Soc. (JEMS
Kan subdivision and products of simplicial sets
The canonical map from the Kan subdivision of a product of finite simplicial
sets to the product of the Kan subdivisions is a simple map, in the sense that
its geometric realization has contractible point inverses
Cubical and cosimplicial descent
We prove that algebraic K-theory, topological Hochschild homology and
topological cyclic homology satisfy cubical and cosimplicial descent at
connective structured ring spectra along 1-connected maps of such ring spectra
The topological Singer construction
We study the continuous (co-)homology of towers of spectra, with emphasis on
a tower with homotopy inverse limit the Tate construction X^{tG} on a
G-spectrum X. When G=C_p is cyclic of prime order and X=B^p is the p-th smash
power of a bounded below spectrum B with H_*(B) of finite type, we prove that
(B^p)^{tC_p} is a topological model for the Singer construction R_+(H^*(B)) on
H^*(B). There is a map epsilon_B : B --> (B^p)^{tC_p} inducing the
Ext_A-equivalence epsilon : R_+(H^*(B)) --> H^*(B). Hence epsilon_B and the
canonical map Gamma : (B^p)^{C_p} --> (B^p)^{hC_p} are p-adic equivalences
- …