On the higher universal quadratic functors and related computations (Q1175757)

In earlier work [J. Algebra 127, No. 1, 178-181 (1989; Zbl 0687.20043)], the author proved the existence of a long exact sequence \[ \to H_{n+1}(G)\to \Gamma_ n(G)\to J_ n(G)\to H_ n(G)\to \dots\to H_ 2(G)\to 0. \] Here \(H_ n(G)\) is the \(n\)th integral homology group of \(G\), \(\Gamma_ 2(G)\) is \(\Gamma(G^{Ab})\), the result obtained by applying Whitehead's universal quadratic functor to the abelianised group, \(G^{Ab}\), and \(J_ 2(G)=\ker(G\otimes G\to G)\). The functors \(\Gamma_ n\) and \(J_ n\) are derived functors of \(\Gamma_ 2\) and -- \(\otimes\) -- respectively. The aim of this paper is to develop methods of calculation for these functors \(\Gamma_ n\) and \(J_ n\). Results relating to their behaviour on extensions, on cyclic groups and on free products are given.