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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Universal Toda brackets of ring spectra


Author: Steffen Sagave
Journal: Trans. Amer. Math. Soc. 360 (2008), 2767-2808
MSC (2000): Primary 55P43; Secondary 19D55, 55S35, 55U35
Published electronically: December 11, 2007
MathSciNet review: 2373333
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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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Additional Information

Steffen Sagave
Affiliation: Department of Mathematics, University of Oslo, Box 1053, N-0316 Oslo, Norway
Email: sagave@math.uio.no

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04487-X
PII: S 0002-9947(07)04487-X
Received by editor(s): December 5, 2006
Published electronically: December 11, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.