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Derived functors of nonadditive functors and homotopy theory

Lawrence Breen and Roman Mikhailov

Algebraic & Geometric Topology 11 (2011) 327–415

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(A,n), 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 πi(M(A,n)) for small values of both i and n.

nonadditive derived functor, Moore space
Mathematical Subject Classification 2000
Primary: 18G55, 18G10
Secondary: 54E30, 55Q40
Received: 18 January 2010
Accepted: 2 August 2010
Published: 8 January 2011
Lawrence Breen
Laboratoire CNRS LAGA
Universite Paris 13
99, avenue Jean-Baptiste Clement
93430 Villetaneuse
Roman Mikhailov
Department of Algebra
Steklov Mathematical Institute
Gubkina 8