Fibrewise localization and completion

Abstract

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.

Topological Hochschild homology of Thom spectra which are E ∞ -ring spectra

“…Rather than an augmentation over F , we will in this setting require augmentation over the appropriate "p-local" or "pcomplete" analogue. We rely on the careful treatment of fiberwise localization and completion given by May [25]. [25].…”

“…We rely on the careful treatment of fiberwise localization and completion given by May [25]. [25]. Note that we must use continuous versions of localization and completion in order to ensure we have continuous functors [15].…”

“…For oriented bundles, there is a Thom isomorphism with Z (p) or Z ∧ p respectively and for unoriented bundles there is a Z/p Thom isomorphism [25]. Now, π 1 (BF (p) ) is the group of p-local units Z × p .…”

“…We rely on the careful treatment of fiberwise localization and completion given by May [25]. Here BF (p) (V ) classifies spherical fibrations with fiber S V (p) and B(F ) ∧ p (V ) classifies spherical fibrations with fiber (S V ) ∧ p (see [25]). Note that we must use continuous versions of localization and completion in order to ensure that we have continuous functors [15].…”

Homotopy pullback of An-spaces and its applications to An-types of gauge groups

“…The finiteness condition for G is used as follows. May [May80,Theorem 4.1] proved that λ * : π r (M 1 (G)) → π r (M 1 (G P )) is a P-localization for r ≥ 2 when G is a finite complex. By the fibration Ω n−2 Map 0 (G ∧n , G) −→ M n (G) −→ M n−1 (G) and the fact that Map 0 (G ∧n , G) → Map 0 (G ∧n P , G P ) P-localizes each connected components for a connected finite complex G [HMR75, Theorem 3.11], one can inductively show that λ * : π r (M n (G)) → π r (M n (G P )) is a P-localization of the abelian group π r (M n (G)) if G is homotopy equivalent to a finite complex and r ≥ 2.…”