Fibrewise Localization and Completion
نویسنده
چکیده
The behavior of fibrewise localization and completion on the classifying space level is analyzed. The relationship of these constructions to fibrewise joins and smash products and to orientations of spherical fibrations is also analyzed. This theory is essential to validate Sullivan's proof of the Adams conjecture. In Sullivan's beautiful proof of the Adams conjecture [20], perhaps the crucial technical point is the behavior on the classifying space level of fibrewise localization and completion. The idea is that, for a spherical fibration Sn+E +B, one can construct new fibrations s;+E{+B and $ ; t + l ? J + ~ . Here T denotes a set of primes, X, and ,fT denote the localization and completion of a (nilpotent) space X at T, and E; and I?$denote "fibrewise" localized and completed total spaces. This procedure gives the following diagram of representable functors on the category W of spaces of the homotopy type of CW-complexes:
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