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Transactions of the American Mathematical Society

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



Hopf constructions and higher projective planes for iterated loop spaces

Authors: Nicholas J. Kuhn, Michael Slack and Frank Williams
Journal: Trans. Amer. Math. Soc. 347 (1995), 1201-1238
MSC: Primary 55P35; Secondary 55P45, 55P47, 55S12
MathSciNet review: 1282890
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Abstract: 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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