A. K. Bousfield

Homotopy limits, completions and localizations

Completions and localizations.- The R-completion of a space.- Fibre lemmas.- Tower lemmas.- An R-completion of groups and its relation to the R-completion of spaces.- R-localizations of nilpotent spaces.- p-completions of nilpotent spaces.- A glimpse at the R-completion of non-nilpotent spaces.- Towers of fibrations, cosimplicial spaces and homotopy limits.- Simplicial sets and topological spaces.- Towers of fibrations.- Cosimplicial spaces.- Homotopy inverse limits.- Homotopy direct limits.- Errata.- Erratum to: The R-completion of a space.- Erratum to: Tower lemmas.- Erratum to: p-completions of nilpotent spaces.

The localization of spectra with respect to homology

IN [8] WE studied localizations of spaces with respect to homology, and we now develop the analogous stable theory. Let Ho” denote the stable homotopy category of CW-spectra. We show that each spectrum E E Ho” gives rise to a natural E*localization functor ( )E: Ho” -+HoS and n : 1 +( )E. For A E Ho”, v:A-+AE is the terminal example of an E*-equivalence going out of A in Ho”. After proving the existence of ES-localizations, we develop their basic properties and discuss in detail the cases where E is connective and E = K. Using EJocalization theory, we obtain results on the convergence of the E*-Adams spectral sequence, and on the class A(Ho”) of acyclicity types of spectra, see [9]. The paper is sectioned as follows: 91. E*-localizations and E*-acyclizations of spectra; 42. Localizations with respect to Moore spectra; 93. General examples of E*-localizations; 34. K-theoretic localizations of spectra; 05. The E*-Adams spectral sequence and the E-nilpotent completion; 06. Convergence theorems for the E*Adams spectral sequence; P7. Duality in &Ho”). E*-localizations of spectra were previously studied by Adams, who sketched a useful outline of the subject in 141. However, the first complete existence proof for such localizations was obtained by the author as an immediate corrollary of work on EJocalizations of spaces, see [8]. The relevant proofs in [81 may be stabilized by using Kan’s semi-simplicial spectra in place of simplicial sets. That approach also shows the existence of “ES-factorizations” for maps of spectra and leads to a Quillen model category framework for “stable homotopy theory with respect to Es.” To make the present exposition more understandable we omit these topics and work in the homotopy category of CW-spectra. This paper grew out of an attempt to understand Doug Ravenel’s work on localizations of spectra with respect to certain periodic homology theories [17], and we are indebted to him for explaining his ideas. We are also indebted to Mark Mahowald and Haynes Miller for supplying the homotopy theoretic theorem underlying our approach to K*-localizations, and we have been significantly influenced by Guido Mislin’s work on K*-localizations of spaces [ 161. We understand that various results in this paper, particularly in 54, were obtained by Frank Adams and David Baird in earlier unpublished work, and that some have been obtained independently by W. Dwyer (unpublished). We essentially use the notation and terminology of [4]. However, we let Ho” be the category of CW-spectra and homotopy classes of maps of degree 0, see [4], p. 144. Thus HO” is an additive category equipped with an equivalence C : Ho” --f Ho” induced by the “shift” suspension for CW-spectra. We let [X, Y] be the group of morphisms

Homotopy spectral sequences and obstructions

For a pointed cosimplicial spaceX•, the author and Kan developed a spectral sequence abutting to the homotopy of the total space TotX•. In this paper,X• is allowed to be unpointed and the spectral sequence is extended to include terms of negative total dimension. Improved convergence results are obtained, and a very general homotopy obstruction theory is developed with higher order obstructions belonging to spectral sequence terms. This applies, for example, to the classical homotopy spectral sequence and obstruction theory for an unpointed mapping space, as well as to the corresponding unstable Adams spectral sequence and associated obstruction theory, which are presented here.

Cosimplicial resolutions and homotopy spectral sequences in model categories

We develop a general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model categories, extending work of Bouseld{Kan and Bendersky{Thompson for ordinary spaces. This is based on a generalized cosimplicial version of the Dwyer{Kan{Stover theory of resolution model categories, and we are able to construct our homotopy spectral sequences and completions using very flexible weak resolutions in the spirit of relative homological algebra. We deduce that our completion functors have triple structures and preserve certain ber squares up to homotopy. We also deduce that the Bendersky{Thompson completions over connective ring spectra are equivalent to Bouseld{Kan completions over solid rings. The present work allows us to show, in a subsequent paper, that the homotopy spectral sequences over arbitrary ring spectra have well-behaved composition pairings.

On the telescopic homotopy theory of spaces

In telescopic homotopy theory, a space or spectrum X is approximated by a tower of localizations LnX, n ≥ 0, taking account of vn-periodic homotopy groups for progressively higher n. For each n ≥ 1, we construct a telescopic Kuhn functor Φn carrying a space to a spectrum with the same vn-periodic homotopy groups, and we construct a new functor Θn left adjoint to Φn. Using these functors, we show that the nth stable monocular homotopy category (comprising the nth fibers of stable telescopic towers) embeds as a retract of the nth unstable monocular homotopy category in two ways: one giving infinite loop spaces and the other giving “infinite Ln-suspension spaces.” We deduce that Ravenel’s stable telescope conjectures are equivalent to unstable telescope conjectures. In particular, we show that the failure of Ravenel’s nth stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial Johnson-Wilson E(n)∗-homology but nontrivial vn-periodic homotopy groups, showing a fundamental difference between the unstable chromatic and telescopic theories. As a stable chromatic application, we show that each spectrum is K(n)-equivalent to a suspension spectrum. As an unstable chromatic application, we determine the E(n)∗-localizations and K(n)∗-localizations of infinite loop spaces in terms of E(n)∗-localizations of spectra under suitable conditions. We also determine the E(n)∗-localizations and K(n)∗-localizations of arbitrary Postnikov H-spaces.

A classification of K-local spectra

Abstract When the stable homotopy category is localized with respect to ordinary topological K -theory, it becomes highly algebraic. In this paper, an algebraic classification of K ∗ -local spectra is obtained using a ‘united K -homology theory’ K CRT ∗ which combines the complex, real, and self- conjugate theories. It has much better homological algebraic properties than its constituent homology theories and leads to a K CRT ∗ -Adams spectral sequence for K ∗ -local mapping class groups which always vanishes above homological degree 2. The main classification results of this paper hold without arithmetic localization and generalize results previously obtained at an odd prime.

A second quadrant homotopy spectral sequence

For each cosimplicial simplicial set with basepoint, the authors construct a homotopv stectral sequence generalizing the usual spectral sequence for a second quadrant double chain complex. For such homotopy spectral sequences, a uniqueness theorem and a general multiplicative pairing are established. This machinery is used elsewhere to show the equivalence of various unstable Adams spectral sequences and to construct for them certain composition pairings and Whitehead products.

THE K-THEORY LOCALIZATIONS AND v1-PERIODIC HOMOTOPY GROUPS OF H-SPACES

Abstract We determine the mod p K -theory localizations and v 1 -periodic homotopy groups of finite H -spaces and of other spaces with torsion-free exterior p -adic K -cohomology algebras at an odd prime p . Our localization results generalize those of Mahowald and Thompson (Topology 1992, 31 , 133–141) for odd-dimensional spheres. We construct our mod p K -theory localizations as homotopy fibers of unstable maps between infinite loop spaces, and similarly construct a wide array of new spaces having torsion-free exterior p -adic K -cohomology algebras with prescribed Adams operations. This leads, for example, to a classification of the odd mod p K -homology spheres.

Unstable localization and periodicity

In the 1980’s, remarkable advances were made by Ravenel, Hopkins, Devinatz, and Smith toward a global understanding of stable homotopy theory, showing that some major features arise “chromatically” from an interplay of periodic phenomena arranged in a hierarchy (see [20], [21], [28]). We would like very much to achieve a similar understanding in unstable homotopy theory and shall describe some progress in that direction. In particular, we shall explain and extend some results of our papers [4], [11], and some closely related results of Dror Farjoun and Smith [17], [18], [19].

On the

This paper deals with the p-adic completion F p ∞X developed by Bousfield-Kan for a space X and prime p. A space X is called F p -good when the map X→F p ∞X is a mod-p homology equivalence, and called F p -bad otherwise. General examples of F p -good spaces are established beyond the usual nilpotent or virtually nilpotent ones. These include the polycyclic-by-finite spaces. However, the wedge of a circle with a sphere of positive dimension is shown to be F p -bad. This provides the first example of an F p -bad space of finite type and implies that the p-profinite completion of a free group on two generators must have nontrivial higher mod-p homology as a discrete group