Iterates of the suspension map and Mitchell’s finite spectra with $A_k$-free cohomology

We study certain cross-effects of the unstable homotopy of spheres. These cross-effects were constructed by Weiss, for different purposes, in the context of “Orthogonal calculus”. We show that Mithchell’s finite spectra with $A_k$-free cohomology (constructed in \cite{mitchell}) arise naturally as stabilizations of Weiss' cross-effects. Moreover, we find that after a suitable Bousfield localization, our cross-effects, which capture meaningful information about the unstable homotopy of spheres, are homotopy equivalent to the infinite loop spaces associated with Mitchell’s spectra. This last result is closely related to the main result of Mahowald and the author in \cite{AM97}.

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Published 1 January 1998