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

Let T(j) be the dual of the j th stable summand of Ω2 S 3 (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.