INTERNAL q-HOMOLOGY OF CROSSED MODULES

[摘要] For $q$ a non-negative integer, we introduce the internal $q$-homology of crossed modules and we obtain in the case $q=0$ the homology of crossed modules. In the particular case of considering a group as a crossed module we obtain that its internal $q$-homology is the homology of the group with coefficients in the ring of the integers modulo $q$.The second internal $q$-homology of crossed modules coincides with the invariant introduced by Grandjeán and López, that is, the kernel of the universal $q$-central extension. Finally, we relate the internal $q$-homology of a crossed module to the homology of its classifying space with coefficients in the ring of the integers modulo $q$.AMS 2000 Mathematics subject classification: Primary 18G50; 20J05. Secondary 18G30

[学科分类] 数学（综合）

[关键词] crossed module;classifying space;q-homology;q-commutator;q-centre