The equivalence of fiber spaces and bundles

Fadell, Edward

doi:10.1090/s0002-9904-1960-10389-8

Cited by 7 publications

(2 citation statements)

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“…If f and £ are fibre bundles, then the constructions in James [11] reduce the question to one about sections of certain fibrations, and the result follows by standard obstruction theory. Use of Fadell's theorem [7] then reduces the general case to the bundle case.…”

Section: Sxmentioning

confidence: 99%

Fibrewise localization and completion

May¹

1980

Trans. Amer. Math. Soc.

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Abstract. 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 S" -> E^> B, one can construct new fibrations SÇ.->E£^B and S} -* J£ -> B.Here T denotes a set of primes, XT and XT denote the localization and completion of a (nilpotent) space X at T, and EST and E{-denote "fibrewise" localized and completed total spaces. This procedure gives the following diagram of representable functors on the category ^ of spaces of the homotopy type of CW-complexes:Here SX(B) is the set of equivalence classes of fibrations with fibre X over a space fief.There results a corresponding diagram of classifying spaces, and the point is to analyze its behavior in terms of localization and completion.One can construct fibrewise localizations and completions in at least four ways. Sullivan [20] takes B to be a simplicial complex and proceeds cell by cell. Bousfield and Kan [3] work with simplicial sets and give a nice functorial construction. It is also possible to work with Postnikov towers and proceed cocell by cocell [16]. None of these procedures bears any obvious relationship to behavior on the classifying space level. The question is not considered by Bousfield and Kan. I do not know whether or not Sullivan's outline of a proof of the relationship could be made rigorous. Certainly any such argument would be extremely long and technical.We shall perform certain natural constructions on the classifying space level. These will give sufficient functoriality (in the variable X) on the classifying spaces for S X as to allow a new construction of fibrewise localizations and completions AMS (MOS) subject classifications (1970). Primary 55E50, 55F05, 55F15.

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Section: Sxmentioning

confidence: 99%

Fibrewise localization and completion

May¹

1980

Trans. Amer. Math. Soc.

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“…By the theorem of [8], the fibration is fiber homotopy equivalent to a bundle. Since orientability is preserved by fiber homotopy equivalence, we may assume without loss of generality that p : E → B is an orientable bundle, with fiber Y .…”

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confidence: 99%

Nielsen numbers of n-valued fiber maps

Brown

2008

J. fixed point theory appl.

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The Nielsen number for n-valued multimaps, defined by Schirmer, has been calculated only for the circle. A concept of n-valued fiber map on the total space of a fibration is introduced. A formula for the Nielsen numbers of n-valued fiber maps of fibrations over the circle reduces the calculation to the computation of Nielsen numbers of single-valued maps. If the fibration is orientable, the product formula for single-valued fiber maps fails to generalize, but a "semi-product formula" is obtained. In this way, the class of n-valued multimaps for which the Nielsen number can be computed is substantially enlarged.

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Spherical fibrations and manifolds

Wissemann-Hartmann

1981

Math Z

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The equivalence of fiber spaces and bundles

Cited by 7 publications

References 7 publications

Fibrewise localization and completion

Fibrewise localization and completion

Nielsen numbers of n-valued fiber maps

Spherical fibrations and manifolds

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