Universal Toda brackets of ring spectra

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

References in corpus (2)

• Universal Toda brackets of ring spectra
• Postnikov extensions of ring spectra

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• Transchromatic extensions in motivic and Real bordism
• The higher structure of unstable homotopy groups
• Realising ${\pi }_{\ast }^{e}R$-algebras by global ring spectra
• Coherent cochain complexes and Beilinson t-structures, with an appendix by Achim Krause
• The bivariant parasimplicial ${\mathsf{S}}_{\bullet }$-construction