An algebraic derivation of a certain exact sequence
\textit{R. Brown}, \textit{D. L. Johnson} and \textit{E. F. Robertson} [J. Algebra 111, 177-202 (1987; Zbl 0626.20038)] have used topological methods to obtain the exact sequence \[ H_ 3(G)\to \Gamma (G^{ab})\to J_ 2(G)\to H_ 2(G)\to 0. \] Here \(H_ i(G)\) denotes the i-th integral homomology group of G, \(J_ 2(G)\) is the (abelian) kernel of the canonical homomorphism from the nonabelian tensor square \(G\otimes G\) to G, and \(\Gamma (G^{ab})\) is the so called ``universal quadratic construction'' due to J. H. C. Whitehead. In this paper the author proceeds in an algebraic way, using methods of categorical homology theory (due to Barr-Beck, a.o.). He shows that the above sequence is the right hand end of a long exact sequence involving the integral homology and the derived functors of \(\Gamma\) and \(\otimes\).
scientific article; zbMATH DE number 4125694
scientific article