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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. Received by the editors February 23, 1979. A M S ( M O S ) subject classificatio (1970). Primary 55E50, 55F05, 55F15.doi:10.1090/s0002-9947-1980-0554323-8 fatcat:pgssfonygbgfrceplphdkohyfq