Universal Toda brackets of ring spectra
arXiv:math/0611808 · doi:10.1090/S0002-9947-07-04487-X
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.
38 pages; a few typos corrected, to appear in Trans. Amer. Math. Soc