Derived functors of nonadditive functors and homotopy theory

Abstract

The main purpose of this paper is to extend our knowledge of the derived functors of certain basic nonadditive functors. The discussion takes place over the integers, and includes a functorial description of the derived functors of certain Lie functors, as well as that of the main cubical functors. We also present a functorial approach to the study of the homotopy groups of spheres and of Moore spaces $M\left(A,n\right)$, based on the Curtis spectral sequence and the decomposition of Lie functors as iterates of simpler functors such as the symmetric or exterior algebra functors. As an illustration, we retrieve in a purely algebraic manner the $3$–torsion components of the homotopy groups of the $2$–sphere in low degrees, and give a unified presentation of the homotopy groups ${\pi }_i\left(M\left(A,n\right)\right)$ for small values of both $i$ and $n$.