Discrete & Computational Geometry

, Volume 53, Issue 1, pp 1–15

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

• Ana Romero
• Francis Sergeraert
Article

## Abstract

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.

## Keywords

Cradle Theorem Discrete vector fields  Effective homotopy

## Notes

### Acknowledgments

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

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