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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
DOI: https://doi.org/10.1090/S0002-9947-1995-1282890-9
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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