A van Kampen spectral sequence for higher homotopy groups
The homotopy groups \(\pi_*(X)\) of a space X, together with the action of all primary homotopy operations (composition, Whitehead product, action of \(\pi_ 1\) etc.) build an algebraic structure called \(\Pi\)- algebra. The category of \(\Pi\)-algebras admits a coproduct functor \(- \coprod -\) and, moreover, \(-\coprod -\) has derived functors in the sense of \textit{D. G. Quillen} [Homotopical algebra (1967; Zbl 0168.209)]. These derived functors applied to \((\pi_*(X),\pi_*(Y))\) yield the \(E^ 2\)- term of a spectral sequence converging to \(\pi_*(X\vee Y)\). More generally, there is a spectral sequence converging to \(\pi_*(_{\to}\underline X)\) where \b{X} is a small diagram of spaces. The spectral sequences are constructed using simplicial resolutions of spaces. The paper contains an appendix on the tensor product of \(\Pi\)- algebras.
scientific article; zbMATH DE number 4140816