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.
