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.
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Notes
An existence theorem is constructive if an explicit construction, in other words an algorithm, is given to exhibit the object the existence of which is claimed. The practical usefulness of this algorithm depends on its complexity. With respect to some dimension, we are quickly here in front of the P-NP problem. On the contrary, in some fixed dimension, the dependency with respect to some initial combinatorial object is polynomial. The previous work of the authors, see for example [3], illustrates the practical interest of these constructive methods.
Pedro Real’s algorithm computing the homotopy groups [5] uses the Whitehead tower, requiring an iterative use of Serre and Eilenberg–Moore spectral sequences, while Bousfield–Kan produces a unique spectral sequence, with a rich algebraic structure, leading in particular, if localized at a prime, to the module structure with respect to the corresponding Steenrod algebra, that is, the Adams spectral sequence.
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Acknowledgments
The research was partially supported by Ministerio de Economía y Competitividad, Spain, project MTM2013-41775-P.
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Romero, A., Sergeraert, F. A Combinatorial Tool for Computing the Effective Homotopy of Iterated Loop Spaces. Discrete Comput Geom 53, 1–15 (2015). https://doi.org/10.1007/s00454-014-9650-1
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DOI: https://doi.org/10.1007/s00454-014-9650-1