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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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


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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Additional Information
  • Steffen Sagave
  • Affiliation: Department of Mathematics, University of Oslo, Box 1053, N-0316 Oslo, Norway
  • Email:
  • Received by editor(s): December 5, 2006
  • Published electronically: December 11, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 2767-2808
  • MSC (2000): Primary 55P43; Secondary 19D55, 55S35, 55U35
  • DOI:
  • MathSciNet review: 2373333