{'fraction_non_alphanumeric': 0.03137789904502047, 'fraction_numerical': 0.0, 'mean_word_length': 4.518796992481203, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Universal Bundles and Classifying Spaces

This chapter evaluates universal bundles and classifying spaces. As before, G is a topological group. In defining the equivariant cohomology of a G-space M, one needs a weakly contractible space EG on which G acts freely. Such a space is provided by the total space of a universal G-bundle, a bundle from which every principal G-bundle can be pulled back. The base BG of a universal G-bundle is called a classifying space for G. By Whitehead's theorem, for CW-complexes, weakly contractible is the same as contractible. In the category of CW complexes (with continuous maps as morphisms), a principal G-bundle whose total space is contractible turns out to be precisely a universal G-bundle.