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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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Hopf constructions and higher projective planes for iterated loop spaces
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by Nicholas J. Kuhn, Michael Slack and Frank Williams PDF
Trans. Amer. Math. Soc. 347 (1995), 1201-1238 Request permission


We define a category, $\mathcal {H}_p^n$ (for each $n$ and $p$), of spaces with strong homotopy commutativity properties. These spaces have just enough structure to define the $\bmod p$ Dyer-Lashof operations for $n$-fold loop spaces. The category $\mathcal {H}_p^n$ is very convenient for applications since its objects and morphisms are defined in a homotopy invariant way. We then define a functor, $P_p^n$, from $\mathcal {H}_p^n$ to the homotopy category of spaces and show $P_p^n$ to be left adjoint to the $n$-fold loop space functor. We then show how one can exploit this adjointness in cohomological calculations to yield new results about iterated loop spaces.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 1201-1238
  • MSC: Primary 55P35; Secondary 55P45, 55P47, 55S12
  • DOI:
  • MathSciNet review: 1282890