Computing the first stages of the Bousfield-Kan spectral sequence (Q969620)

The article is devoted to the calculation of the first two stages of the Bousfield-Kan spectral sequence. The Bousfield-Kan spectral sequence presents the Adam spectral sequence in the setting of combinatorial topology and makes it the major tool for computation of homotopy groups. In the article the author makes an attempt to use the effective homotopy method to construct algorithms for computing the Bousfield-Kan spectral sequence. The author develops an effective homology version of the constructor associating to every simplicial set \(X\) the free Abelian group \(RX\). The constructor is used to obtain the first two levels of the Bousfield-Kan spectral sequence, the ``pages'' \(E^1\) and \(E^2\). We have to notice that the general algorithm computing the higher levels is still under work. We present some main results. Theorem 3. Let \(X\) be a 1-reduced pointed simplicial set, and \(E=(E^r,d^r)_{r\geq1}\) the associated Bousfield-Kan spectral sequence. Then \(E^1\) satisfies \[ E^1_{p,q}=0\quad\text{if }q<2p+2. \] Algorithm I. Input: 1) a 1-reduced pointed simplicial set \(X\), 2) an equivalence \(C_*(X)\Leftarrow DX_*\Rightarrow HX_*\), where \(HX_*\) is an effective chain complex. Output: an equivalence \(\mu_L:C_*(RX)\Leftarrow C_*(\Gamma(\widetilde{DX_*}))\Rightarrow C_*(\Gamma(\widetilde{HX_*}))\), where \(\widetilde{DX_*},\widetilde{HX_*}\) are obtained from \(DX_*, HX_*\) respectively, \(\widetilde{HX_*}\) is effective and \(\widetilde{HX_0}=\widetilde{HX_1}=0\).