New Relationships among Loopspaces, Symmetric Products, and Eilenberg Maclane Spaces

Let T (j) be the dual of the jth stable summand of Ω2S3 (at the prime 2) with top class in dimension j. Then it is known that T (j) is a retract of a suspension spectrum, and that the homotopy colimit of a certain sequence T (j) → T (2j) → . . . is an infinite wedge of stable summands of K(V, 1)’s, where V denotes an elementary abelian 2 group. In particular, when one starts with T (1), one gets K(Z/2, 1) = RP∞ as one of the summands. I discuss a generalization of this picture using higher iterated loopspaces and Eilenberg MacLane spaces. I consider certain finite spectra T (n, j) for n, j ≥ 0 (with T (1, j) = T (j)), dual to summands of Ωn+1SN , conjecture generalizations of the above, and prove that these conjectures are correct in cohomology. So, for example, T (n, j) has unstable cohomology, and the cohomology of the hocolimit of a certain sequence T (n, j) → T (n, 2j) → . . . agrees with the cohomology of the wedge of stable summands of K(V, n)’s corresponding to the wedge occurring in the n = 1 case above. One can also map the T (n, j) to each other as n varies, and here the cohomological calculations imply a homotopical conclusion: the hocolimits that are nonzero, T (∞, 2k), for k ≥ 0, map to each other, giving rise to a fitration of HZ/2 which is equivalent to the mod 2 symmetric powers of spheres filtration. Our homotopical constructions use Hopf invariant methods and loopspace technology. These are quite general and should be of independent interest. To study the action of the Steenrod operations on the cohomology of our spectra, we derive a Nishida formula for how χ(Sqi) acts on Dyer–Lashof operations. This should be of use in other settings. In an appendix, we explain connections with recent work by Greg Arone and Mark Mahowald on the Goodwillie tower of the identity.