On the homotopy theory of simplicial Lie algebras

Abstract

Elements X,, n 0, which generate the homotopy groups of spheres in the category of simplicial Lie algebras are shown to have Hopf invariant one. This fact is shown to have strong implications for the hoinotopy theory of this category. In 1958, Kan [5] constructed an algebraic model (simplicial groups) for homotopy theory. Since then, various group theoretic methods liave been used to study this model. In 1965, Curtis [3] showed that the lower central series filtration of a group induces a spectral sequence for computing homotopy groups of a simplicial group. This spectral sequence starts with the homotopy groups of a simplicial Lie algebra. In this sense the homotopy theory of simplicial Lie algebras is a first approximation to ordinary hornotopy theory. The purpose of this note is to describe this approximation from the point of view of an appropiiately defined analogue of the Hopf invarianit. We shall be concerned with the followving statements which reveal something of the simplicity of the loomotopy structure of simplicial Lie algebras. A. There are elements of Hopf invariant one for every integer n?O. B. The Steenrod algebra is bigraded with Sqi having bidegree (i, 1) for i> O; SqO is identically zero. C. The Adams spectral sequence for splheres collapses (E2=E00). D. The homotopy groups of spheres are generated by elements of Hopf invariant one under composition. E. The EHP sequence is short exact. This note is intended as an epilogue to [I ] in which a stable mod p version of the Curtis spectral sequence yielding a newv (E1, d1)-term of the Adams spectral sequence is studied.' Results A-E are due to the authors of [1] and the present author. All have appeared previously [I], [9], [4a], with the exception of A, which was first noted by D. Quillen. We hope, however, that the general homotopy theorist will find the Hopf invariant theme used here more intuitive. Received by tbe editors July 30, 1969. A MS Subject Classifications. Primary 5540, 5534.