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

This paper is devoted to the Cradle Theorem. It is a combinatorial contraction discovered when studying a crucial point of the effective Bousfield-Kan spectral sequence, an unavoidable step to make effective the famous Adams spectral sequence. The homotopy equivalence TOP(Sp,TOP(Sq, X)) ∼ TOP(Sp+q, X) is obvious in ordinary topology, not surprising in combinatorial topology, but it happens the proof in the last case is relatively difficult, it is essentially our Cradle Theorem. Based on a simple and natural discrete vector field, it produces also new tools to understand and efficiently implement the Eilenberg-Zilber theorem, the usual one and the twisted one as well. Once the Cradle Theorem is proved, we quickly explain its role in the Bousfield-Kan spectral sequence, via the notion of effective homotopy. An interested reader must know the present paper is elementary, in particular no knowledge of the Bousfield-Kan and Adams spectral sequences is required. The cradle theorem could also be a good opportunity to enter the subject of discrete vector fields and to understand its power in an unexpected field: the structure of simplicial products.