Universal Toda brackets of ring spectra

Abstract

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