A generalized Hopf formula for higher homology groups
Let G be a discrete group, let \(\{1\to R_ i\to F_ i\to G\to 1\}_{i=1,...,c}\) be a family of free presentations of G, let F be the free product \(F=F_ 1*...*F_ c\), and, for each \(1\leq i\leq c\), identify \(F_ i\) and \(R_ i\) with their images in F. Further, for any group H, write \(H=\gamma_ 1H\supseteq \gamma_ 2H\supseteq..\). for its lower central series. In the paper an isomorphism between the integral homology group \(H_{2c}(G)\) and the group \[ ([R_ 1,...,R_ c]\cap N)\gamma_{c+1}R/[R_ 1,...,R_ c,F]\gamma_{c+1}R \] is established where R and N are suitably defined normal subgroups of F. For \(c=1\) this boils down to the usual Schur-Hopf formula. \{Reviewer's remark. The exact sequence (2) in the paper attributed to Gruenberg and Blackburn may be deduced from results of \textit{H. G. Schumann} [Math. Ann. 114, 385-413 (1937; Zbl 0016.29403)].\}
scientific article; zbMATH DE number 4122232