New relationships among loopspaces, symmetric products, and Eilenberg MacLane spaces

Abstract. 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 innite 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) = RP1 as one of the summands. I discuss a generalization of this picture using higher iterated loopspaces and Eilenberg MacLane spaces. I consider certain nite 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 co-homology 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 coho