Discrete & Computational Geometry

, Volume 53, Issue 1, pp 1–15 | Cite as

A Combinatorial Tool for Computing the Effective Homotopy of Iterated Loop Spaces

  • Ana RomeroEmail author
  • Francis Sergeraert


This paper is devoted to the Cradle Theorem. It is a recursive description of a discrete vector field on the direct product of simplices \(\varDelta ^p \times \varDelta ^q\) endowed with the standard triangulation. The vector field provides an explicit deformation that is used to establish an algorithm for computing the Bousfield–Kan spectral sequence, more precisely to compute the homotopy groups \(\pi _n(\varOmega ^p G)\) for \(G\) a 1-reduced simplicial abelian group.


Cradle Theorem Discrete vector fields  Effective homotopy 



The research was partially supported by Ministerio de Economía y Competitividad, Spain, project MTM2013-41775-P.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.University of La RiojaLogroñoSpain
  2. 2.Institut FourierGrenobleFrance

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