Universal Toda brackets of ring spectra

Details Universal Toda brackets of ring spectra Universal Toda brackets of ring spectra

2006-11-27

arxiv.org/abs/math/0611808v3

Trans. Amer. Math. Soc., 360(5):2767-2808, 2008.

Mathematics

Algebraic Topology

38 pages; a few typos corrected, to appear in Trans. Amer. Math. Soc

Scientific paper

10.1090/S0002-9947-07-04487-X

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.

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