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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 Akfree cohomology (constructed in [Mt85]) arise naturally as stabilizations of Weiss’ cross-effects. In particular, 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 a partial generalization of the main result of Mahowald and the author in [AM97].

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