A homotopy operation spectral sequence for the computation of homotopy groups
Suppose that \(X\) is an \((r-1)\)-connected space with \(r \geq 2\). The main result of this paper (Theorem 2.4) asserts the existence of a spectral sequence with \(E_{2}\)-term consisting of derived homotopy operations, and which converges to the homotopy groups of the homotopy fibre \(\Gamma X\) of the canonical map \(X \rightarrow S_{\infty}X\) taking values in the infinite symmetric product of \(X\). The spectral sequence is the Bousfield-Friedlander spectral sequence for a model of \(\Gamma X\) in the category of pointed bisimplicial sets. This model is the homotopy fibre of the simplicial set Hurewicz map on an \(E_{2}\)-cofibrant resolution of the space \(X\). The construction of the spectral sequence appears in the last section of the paper, while consequences of the construction are explained in some detail in earlier sections. A description of homotopy functors and their derived functors also appears early in the paper. The spectral sequence is a general computational machine for the homotopy groups of \(\Gamma X\), and its construction completes a calculational program for these groups which was initiated by J.H.C. Whitehead in the early 1950s.
scientific article; zbMATH DE number 1356316