Hopf Formulae for the Higher Homology of a Group

H3(G) « {R, n R2 n [F, F]}/{[F, R, 0^) [Rlt R2]}. \ Note that for any group G and n ^ 1, such an F and Rt can be found: let F (G) be the free group on G; define inductively Ft = F^-^G)), and set F= F (G); for 1 ^ / ^ n let e(: F (G) -> F~\G) denote the canonical homorphisms induced by applying F"'* to the standard 'augmentation' map F^F"^)) ->F~\G) (where /ro((7) = Gyf a n d s e t ^ ( = K e r f i i An alternative method, analogous to methods in [4,5], is best illustrated for n = 2. Choose any surjections F{^G with F{ free, / = 1,2. Let P be the pullback of these surjections and choose a surjection F-» P with F free. Let Rf be the kernel of the composite F-> P -> Ft. In general, one constructs inductively an «-cube of groups F such that, for A c : (i) /r j s free if ^ ^ , (ii) /r j s G for ^ = , and (iii) the morphism FA -»limB=)/4FB is surjective. Such an /i-cube might be called afibrant n-presentation of G. Again, suppose G = F/HKwhere / /and # a r e normal subgroups of Fsuch that F, F/H and F/KSLTQ free. For example, we might be given a presentation