A Bousfield–Kan Algorithm for Computing the Effective Homotopy of a Space

This paper is devoted to a constructive version of the Bousfield–Kan spectral sequence (BKSS). The BKSS provides a combinatorial basis for the famous Adams spectral sequence and its descendants. Its systematic description in the book “Homotopy Limits, Completions and Localizations” remains a relatively difficult text, often cited by its sweet nickname, the “Yellow Monster.” The modern constructive point of view gives an opportunity to reread this essential text and to use it to produce a new algorithm computing homotopy groups, more precisely computing the effective homotopy of a given space, a new concept much richer than the ordinary homotopy groups. Without changing the general philosophy of the BKSS, the constructive constraint leads to a significant reorganization of this rich material and, as it is most often the case, finally to a simpler and more explicit description. Combined with our own basic tools, effective homology and effective homotopy, the description of the BKSS given here is finally not so complicated and could also help the topologists interested by this nice subject.