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Hopf formulae for the higher homology of a group
 Bull. London Math. Soc
, 1988
"... In this note we generalise Hopfs formula for the second homology of a group G in terms of a free presentation R> • F^ » G. We prove: THEOREM 1. Let Rlt...,Rn be normal subgroups of a group F such that F/Y\iiiinRi = G, and for each proper subset A of <«> = {1,...,«} the groups Hr(F/Y[ieA R,) are tri ..."
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In this note we generalise Hopfs formula for the second homology of a group G in terms of a free presentation R> • F^ » G. We prove: THEOREM 1. Let Rlt...,Rn be normal subgroups of a group F such that F/Y\iiiinRi = G, and for each proper subset A of <«> = {1,...,«} the groups Hr(F/Y[ieA R,) are trivial for r = 2ifA = 0, and for r = \A \ + 1 and \A \ + 2 if A # 0 (for example, the groups F/\\ieARi are free for A£(n)). Then there is an isomorphisml)l /»£<«> ieA ilA s tne Here \A \ denotes the order of A, and ^6^1 * subgroup of F generated by trivial subgroup). Also is understood to mean F. Thus for n = 2 the formula reads the subgroups Ri with ieA (in particular, I~[ie0^i * S ^ e f]ie0Rt 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 l (G) be the free group on G; define inductively Ft = F^^G)), and set F = F n (G); for 1 ^ / ^ n let e(: F n (G)> F n ~\G) denote the canonical homorphisms induced by applying F"' * to the standard 'augmentation ' map F^F 1 " 1 ^))>F l ~\G) (where /ro((7) = Gyf and set ^ ( = Kerfii 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 js free if ^ ^ <«>, (ii) /r js G for ^ = <«>, and (iii) the morphism FA»limB=)/4FB is surjective. Such an /icube might be called afibrant npresentation of G. Again, suppose G = F/HKwhere //and #are normal subgroups of Fsuch that F, F/H and F/KSLTQ free. For example, we might be given a presentation <A ^ £/, V) of
A Homotopy 2Groupoid From a Fibration
, 1999
"... In this paper we give an elementary derivation of a 2groupoid from a fibration. This extends a previous result for pointed fibrations due to Loday. Discussion is included as to the translation between 2groupoids and cat 1 groupoids. ..."
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In this paper we give an elementary derivation of a 2groupoid from a fibration. This extends a previous result for pointed fibrations due to Loday. Discussion is included as to the translation between 2groupoids and cat 1 groupoids.