Vol. 60, No. 1, 1975

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On the action of the Dyer-Lashof algebra in H∗(G)

Ib Henning Madsen

Vol. 60 (1975), No. 1, 235–275

Let G be the space of homotopy equivalences of Sn for n →∞. This is an infinite loop space, that is, it has definite deloopings. The first delooping of G is the classifying space for (stable) spherical fibrations.

The ( mod. 2) homology ring of an infinite loop space is an algebra over the Dyer-Lashof algebra R of all primary homology operations. The principal result of this paper is the evaluation of the R-action in H(G). The R-module H(G) determines the R-module H(G∕O), where G∕O is the homogeneous space associated with the infinite orthogonal subgroup of G. Let α : BSO G∕O be a “solution” of the Adams conjecture in the 2-local category, and let QH(G∕O) be the R-module of indecomposable elements.

Mathematical Subject Classification
Primary: 55G25
Secondary: 57F99
Received: 22 January 1974
Published: 1 September 1975
Ib Henning Madsen