Vol. 79, No. 1, 1978

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A spectral sequence for the homology of an infinite delooping

Haynes Miller

Vol. 79 (1978), No. 1, 139–155

Let E = {En : n Z} be a (1)-connected infinite loop space: i.e., ΩEn+1 = En for all n, and for n 0 the space En is (n 1)-connected. Then the stable homology of E is

H∗(E ) = lim H∗+n(En )

under the suspension homomorphisms. One also has the unstable homology H(E0), 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(E0) determines H(E); and the purpose of this paper is to construct and study the general properties of a spectral sequence whose E2-term depends functorially on H(E0) as an R-Hopf algebra and whose E-term is the associated graded module of a natural filtration on H(E). For simplicity we mainly treat the case p = 2.

Mathematical Subject Classification 2000
Primary: 55P47
Secondary: 55S12, 55T99
Received: 24 October 1977
Published: 1 November 1978
Haynes Miller
Massachusetts Institute of Technology
Rm 2-237
Cambridge MA 02139-4307
United States