A spectral sequence for the homology of an infinite delooping

Abstract

Let E = {E n \ n e Z) be a (-l)-connected infinite loop space: i.e., E n+1 = E n for all n, and for w ^ 0 the space E n is (w-l)-connected. Then the stable homology of E is #*(#) = lim H* +n (E n ) under the suspension homomorphims. One also has the unstable homology H*(E 0 ), which with mod p coefficients carries a Pontrjagin product and an action of the mod p Dyer-Lashof algebra R. It is natural to ask how H*(E 0 ) determines H*(E); and the purpose of this paper is to construct and study the general properties of a spectral sequence whose -2 2 -term depends functorially on H*(E 0 ) as an u?-Hopf algebra and whose Eterm is the associated graded module of a natural filtration on H*(E). For simplicity we mainly treat the case p = 2.