Universal Toda brackets of ring spectra

Abstract

Abstract. 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.

Higher Toda brackets and the Adams spectral sequence in triangulated categories

“…We show that our definition agrees with those of [40] and [36]. (We believe that it sometimes differs in sign from [15].…”

“…On the other hand, higher Toda brackets can also be defined in an arbitrary triangulated category. This was done by Shipley in [40], based on Cohen's construction for spaces and spectra [15], and was studied further in [36]. The goal of this paper is to describe precisely how the Adams d r can be described as a particular subset of an (r + 1)-fold Toda bracket which can be viewed as an r th order cohomology operation, all in the context of a general triangulated category.…”

The algebra of secondary homotopy operations in ring spectra

“…represented by π * , * R is the universal Toda bracket [8,53] which also represents the homotopy category of free R-modules as a linear extension of categories, as we prove in Theorem 11.7. The good algebraic properties of our stable secondary homotopy groups make them suitable to model the secondary homotopy of any brave new algebraic structure.…”

Toda Brackets and Cup-One Squares for Ring Spectra

“…One of the definitions uses Toda brackets for triangulated categories in the sense of [Hel68] applied to the homotopy category of R-modules. This is also the definition adopted in [Sag06]. The alternative definition uses tracks, i.e.…”