Abstract
A natural map between group-cohomology of the structure group of a principal fiber bundle with coefficients in the space of functions from the total space into an Abelian group and Čech-cohomology of the base space is defined. A differential complex of local group-cochains is constructed and an analog of the Poincaré lemma for group-cohomology is proven. By using the machinery of spectral sequences the cohomology of this complex is calculated and the connection between group-cohomology and Čech-cohomology of the given principal fiber bundle is elucidated. Finally, the non-Abelian and Witten anomaly in this context is reviewed and the relevance of our results for lifting principal group actions is discussed.
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