Steffen Sagave

Diagram spaces and symmetric spectra

Abstract We present a general homotopical analysis of structured diagram spaces and discuss the relation to symmetric spectra. The main motivating examples are the I -spaces, which are diagrams indexed by finite sets and injections, and J -spaces, which are diagrams indexed by Quillen’s localization construction Σ − 1 Σ on the category Σ of finite sets and bijections. We show that the category of I -spaces provides a convenient model for the homotopy category of spaces in which every E ∞ space can be rectified to a strictly commutative monoid. Similarly, the commutative monoids in the category of J -spaces model graded E ∞ spaces. Using the theory of J -spaces we introduce the graded units of a symmetric ring spectrum. The graded units detect periodicity phenomena in stable homotopy and we show how this can be applied to the theory of topological logarithmic structures.

Group completion and units in I-spaces

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.

Universal Toda brackets of ring spectra

We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an R-module spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex K-theory spectra serve as our main examples.

Spectra of units for periodic ring spectra and group completion of graded E∞ spaces

We construct a new spectrum of units for a commutative symmetric ring spectrum that detects the difference between a periodic ring spectrum and its connective cover. It is augmented over the sphere spectrum. The homotopy cofiber of its augmentation map is a non-connected delooping of the usual spectrum of units whose bottom homotopy group detects periodicity. Our approach builds on the graded variant of E1 spaces introduced in joint work with Christian Schlichtkrull. We construct a group completion model structure for graded E1 spaces and use it to exhibit our spectrum of units functor as a right adjoint on the level of homotopy categories. The resulting group completion functor is an essential tool for studying ring spectra with graded logarithmic structures. 55P43; 55P48

Localization sequences for logarithmic topological Hochschild homology

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

Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories

K

GENERALIZED THOM SPECTRA AND THEIR TOPOLOGICAL HOCHSCHILD HOMOLOGY

K-theory.

On the algebraic K-theory of model categories

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.