N-fold Čech Derived Functors and Generalised Hopf Type Formulas

In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-fold Čech derived functors of the lower central series functors Zk. The paper ends with an application to algebraic K-theory. Introduction and Summary The well known Hopf formula for the second integral homology of a group says that for a given group G there is an isomorphism H2(G) ∼= R ∩ [F, F ] [F,R] , where R F G is a free presentation of the group G. Several alternative generalisations of this classical Hopf formula to higher dimensions were made in various papers, [9, 28, 30], but perhaps the most successful one, giving formulas in all dimensions, was by Brown and Ellis, [3]. They used topological methods, and in particular the Hurewicz theorem for n-cubes of spaces, [5], which itself is an application of the generalised van Kampen theorem for diagrams of spaces [4]. The end result was: 1991 Mathematics Subject Classification. 18G50, 18G10.