Computing the first stages of the Bousfield-Kan spectral sequence

Abstract

In this paper, an algorithm computing the terms E 1 and E 2 of the Bousfield-Kan spectral sequence of a 1-reduced simplicial set X is defined. In order to compute the ordinary description of the first level E 1, some elementary operations of Homological Algebra are sufficient. On the contrary, to compute the stage E 2 it is necessary to know more information about the previous groups, in particular with respect to the generators. This additional information can be reached by computing the effective homology of RX, RX being the free simplicial Abelian group generated by X. The algorithm to get the effective homology of RX from the effective homology of X can be considered the main result in our paper. Moreover, we include a combinatorial proof of the convergence of the Bousfield-Kan spectral sequence, based on the tapered nature of the stage E 1. © 2010 Springer-Verlag.