CCG-homology of crossed modules via classifying spaces
The authors identify the internal homology of a crossed module in terms of the cofiber of certain map. More precisely, a crossed module is a triple \((T,G,\partial)\), where \(\partial\) is a \(G\)-equivariant group homomorphism \(T\to G\), \(G\) acts on \(T\) and the action of \(G\) on \(G\) is by conjugation. The internal homology (\(H_*^{CCG}\)) of the crossed module \((T,G,\partial)\) is defined as the homotopy groups of \((((T,G,\partial)_*)_{ab})\) where the latter is a projective resolution of \((T,G,\partial)\) in the category of crossed modules. By definition this is a homomorphism denoted by \(\zeta H^{CCG}(T,G,\partial)\to\kappa H^{CCG}(T,G,\partial)\). In a previous work of \textit{P. Carrasco, A. M. Cegarra}, and \textit{A. R. Grandjean} [(Co)homology of crossed modules, preprint 1999], the latter has been identified to be isomorphic to the homology of \(G\). The present work identifies \(\zeta H_{i}^{CCG}(T,G,\partial)\) to be isomorphic to \(H_{i+1}(\beta(T,G,\partial))\) where \(\beta(T,G,\partial)\) is defined as the cofiber of the natural map \(i_{(T,G,\partial)}:BG\to B(T,G,\partial)\), here \(B(\cdot)\) stands for the classifying space associated to \((\cdot)\). From this the authors derive the long exact sequence: \(\cdots H_{n+1} (B(T,G,\partial))\to\zeta H^{CCG}_{n}(T,G,\partial)\to H_n(G)\to H_n(B(T,G,\partial))\to \cdots\). Finally, given a \(G\)-module \(M\) and taking the crossed module \((M,G,0)\) the authors identify \(\zeta H_{i+1}^{CCG}(T,G,\partial)\) as isomorphic to \(H_i(G,M)\) and similar long exact sequences as above. There are a few missprints in the paper such as in the definition of \(\beta(T,G,\partial)\), just above Proposition 1. The statement (1) of Corollary 4 is incorrect, it should be \(\zeta H^{CCG}_n(T,G,\partial)\) on the left hand side. On page 662, when identifying \(L(M,2)\), there is a missprint in the Borel construction.
scientific article; zbMATH DE number 1502169