#### Volume 11, issue 1 (2011)

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Derived functors of nonadditive functors and homotopy theory

### Lawrence Breen and Roman Mikhailov

Algebraic & Geometric Topology 11 (2011) 327–415
##### 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 $n$.

##### Keywords
nonadditive derived functor, Moore space
##### Mathematical Subject Classification 2000
Primary: 18G55, 18G10
Secondary: 54E30, 55Q40
##### Publication
Received: 18 January 2010
Accepted: 2 August 2010
Published: 8 January 2011
##### Authors
 Lawrence Breen Laboratoire CNRS LAGA Universite Paris 13 99, avenue Jean-Baptiste Clement 93430 Villetaneuse France http://www.math.univ-paris13.fr/~breen Roman Mikhailov Department of Algebra Steklov Mathematical Institute Gubkina 8 Moscow 119991 Russia http://www.mi.ras.ru/~romanvm/