Universal Toda brackets of ring spectra

Sagave, Steffen

doi:10.1090/S0002-9947-07-04487-X

Universal Toda brackets of ring spectra
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Trans. Amer. Math. Soc. 360 (2008), 2767-2808 Request permission

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.

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