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In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding infinity-categorical product and allows for convenient constructions of commutative parametrized ring spectra. As an immediate application, we compare various models for generalized Thom spectra. In a companion paper, this approach is used to compare homotopical and operator algebraic models for twisted K-theory., v2: 59 pages, exposition improved

We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f: X->BF, where BF denotes the classifying space for stable spherical fibrations. To do this, we consider symmetric monoidal models of the category of spaces over BF and corresponding strong symmetric monoidal Thom spectrum functors. Our main result identifies the topological Hochschild homology as the Thom spectrum of a certain stable bundle over the free loop space L(BX). This leads to explicit calculations of the topological Hochschild homology for a large class of ring spectra, including all of the classical cobordism spectra MO, MSO, MU, etc., and the Eilenberg-Mac Lane spectra HZ/p and HZ., 58 pages

We analyze the equivariant restriction (or transfer) maps in topological Hochschild homology associated to inclusions of group rings of the form R[H]→R[G], where R is a symmetric ring spectrum, G is a discrete group and H⊆G is a subgroup of finite index. This leads to a complete description of the associated restriction (or transfer) maps in topological cyclic homology in terms of the well-known stable transfers in equivariant stable homotopy theory. More generally, we analyze the restriction maps encountered in connection with monoid rings such as polynomial rings and truncated polynomial rings. As a first application of these results we prove a conjecture by Bokstedt, Hsiang and Madsen on how the transfer maps in Waldhausen's algebraic K-theory of spaces relate to the transfers in the stable equivariant homotopy category of a finite cyclic group. As a second application we calculate the subgroup of transfer invariant homotopy classes and we show that the TC-analogue of the lower K-groups vanish below degree −1.

We give a functorial description of the topological cyclic homology of a ring A in terms of the relative algebraic K-theory of the truncated polynomial rings A n = A [ x ] / x n . This description involves the projection and transfer maps relating the relative K-theory spectra K ˜ ( A n ) when n varies. From this point of view the cyclotomic trace corresponds to multiplication by the units 1 + x + ⋯ + x n - 1 in K ˜ 1 ( Z [ x ] / x n ) .

Let GL_1(R) be the units of a commutative ring spectrum R. In this paper we identify the composition BGL_1(R)->K(R)->THH(R)->��^{\infty}(R), where K(R) is the algebraic K-theory and THH(R) the topological Hochschild homology of R. As a corollary we show that classes in ��_{i-1}(R) not annihilated by the stable Hopf map give rise to non-trivial classes in K_i(R) for i\geq 3., Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper16.abs.html

In this paper we continue our study of logarithmic topological Hochschild homology. We show that the inclusion of the connective Adams summand into the p-local complex connective K-theory spectrum, equipped with suitable log structures, is a formally log THH-etale map, and compute the V(1)-homotopy of their logarithmic topological Hochschild homology spectra. As an application, we recover Ausoni's computation of the V(1)-homotopy of the ordinary THH of ku., v3: 32 pages; slightly revised. Accepted for publication in J. Eur. Math. Soc. (JEMS)

We introduce a convenient framework for constructing and analyzing orthogonal Thom spectra arising from virtual vector bundles. This framework enables us to set up a theory of orientations and graded Thom isomorphisms with good multiplicative properties. The theory is applied to the analysis of logarithmic structures on commutative ring spectra., Comment: v3: 39 pages, minor revision. Accepted for publication in Mathematische Zeitschrift

We study the logarithmic topological Hochschild homology of ring spectra with logarithmic structures and establish localization sequences for this theory. Our results apply, for example, to connective covers of periodic ring spectra like real and complex topological K-theory., Comment: v3: 40 pages; minor changes, accepted for publication in Mathematische Annalen

We consider the topological Hochschild homology (THH) of a group ring R[G], and calculate the restriction map (or transfer) associated with a subgroup K ⊆ G of finite index in terms of ordinary group homology transfers. This gives information on the corresponding restriction map in Quillen's K-theory via the topological Dennis trace tr:K(R[G]) → THH(R[G]). More generally, we consider group rings for “rings up to homotopy” (FSP's) and calculate the THH-rcstriction map in terms of transfers in generalized homology theories.

The category of I-spaces is the diagram category of spaces indexed by finite sets and injections. This is a symmetric monoidal category whose commutative monoids model all E-infinity spaces. Working in the category of I-spaces enables us to simplify and strengthen previous work on group completion and units of E-infinity spaces. As an application we clarify the relation to Gamma-spaces and show how the spectrum of units associated with a commutative symmetric ring spectrum arises through a chain of Quillen adjunctions., Comment: v3: 43 pages. Minor revisions, accepted for publication in Algebraic and Geometric Topology

The purpose of this paper is to present a simple and explicit construction of the Bokstedt-Hsiang-Madsen cyclotomic trace relating algebraic K-theory and topological cyclic homology. Our construction also incorporates Goodwillie's idea of a global cyclotomic trace., Comment: 33 pages

In this paper we analyze the higher topological Hochschild homology of commutative Thom S-algebras. This includes the case of the classical cobordism spectra MO, MSO, MU, etc. We consider the homotopy orbits of the torus action on iterated topological Hochschild homology and we describe the relationship to topological Andre-Quillen homology., Comment: 30 pages, minor revisions