Monoidality of Franke?s exotic model

We discuss the monoidal structure on Franke?s algebraic model for the K(p)-local stable homotopy category at odd primes and show that its Picard group is isomorphic to the integers.     

Algebraic models for equivariant homotopy theory over Abelian compact Lie groups

We describe some algebraic models for equivariant rational and p-adic homotopy theory over Abelian compact Lie groups. Received: 12 February 2001; in final form: 15 August 2001 / Published online: 28 February 2002     

Mod 3 cohomology algebras of finite H-spaces

Let X be a 3 local, finite, simply connected H-space with associative homology ring . Some known examples are the Lie group , Harper's H-space X(3) and any odd dimensional sphere . We prove the cohomology algebra is isomorphic to the cohomology algebra of a finite product of and odd dimensional spheres. Received: 15 May 2001; in final form: 22 May 2001 / Published online: 28 February 2002     

On homotopy rigidity of the functor ΣΩ on co-H-spaces

In this paper we study the homotopy rigidity property of the functors ΣΩ and Ω. Our main result is that both functors are homotopy rigid on simply-connected p-local finite co-H-spaces. The result is obtain by a subtle interplay of homotopy decomposition techniques, modular representation theory and the counting principle.     

Realization of 2-solvable nilpotent groups as groups of classes of homotopy self-equivalences

Let X be a 1-connected CW-complex of finite type and ε_?(X) be the group of homotopy classes of self-equivalences of X which induce the identity on homotopy groups. In this paper, we prove that every finitely generated 2-solvable rational nilpotent group is realizable as ε_?(X) where X is the rationalization of a 1-connected CW-complex of finite type.     

Classifying spaces of Kač-Moody groups

We study the structure of classifying spaces of Kač-Moody groups from a homotopy theoretic point of view. They behave in many respects as in the compact Lie group case. The mod p cohomology algebra is noetherian and Lannes'T functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p-subgroups. Also, spaces of maps from classifying spaces of finite p-groups to classifying spaces of Kač-Moody groups are described in terms of classifying spaces of centralizers while the classifying space of a Kač-Moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p-subgroups, up to p-completion. We show that these properties are common to a larger class of groups, also including parabolic subgroups of Kač-Moody groups, and centralizers of finite p-subgroups. Received: 15 June 2000 / in final form: 20 September 2001 / Published online: 29 April 2002     

Models for homotopy n-types in diagram categories

The classical Mac Lane-Whitehead equivalence showing that crossed modules of groups are algebraic models of connected homotopy 2-types has found a corresponding equivariant version by Moerdijk and Svensson ([22]). In this paper we show that this equivariant result has a higher-dimensional version which gives an equivalence between the homotopy category of diagrams of certain objects indexed by the orbit category of a group H and H-equivariant homotopy n-types for n1.Supported by DGICYT:PS90-0226     

The dualizing spectrum of a topological group

To a topological group G, we assign a naive G-spectrum , called the dualizing spectrum of G. When the classifying space BG is finitely dominated, we show that detects Poincaré duality in the sense that BG is a Poincaré duality space if and only if is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behaves multiplicatively on certain topological group extensions. In proving these results we introduce a new tool: a norm map which is defined for any G and for any naive G-spectrum E. Applications of the dualizing spectrum come in two flavors: (i) applications in the theory of Poincaré duality spaces, and (ii) applications in the theory of group cohomology. On the Poincaré duality space side, we derive a homotopy theoretic solution to a problem posed by Wall which says that in a fibration sequence of fini the total space satisfies Poincaré duality if and only if the base and fiber do. The dualizing spectrum can also be used to give an entirely homotopy theoretic construction of the Spivak fibration of a finitely dominated Poincaré duality space. We also include a new proof of Browder's theorem that every finite H-space satisfies Poincaré duality. In connection with group cohomology, we show how to define a variant of Farrell-Tate cohomology for any topological or discrete group G, with coefficients in any naive equivariant cohomology theory E. When E is connective, and when G admits a subgroup H of finite index such that BH is finitely dominated, we show that this cohomology coincides with the ordinary cohomology of G with coefficients in E in degrees greater than the cohomological dimension of H. In an appendix, we identify the homotopy type of for certain kinds of groups. The class includes all compact Lie groups, torsion free arithmetic groups and Bieri-Eckmann duality groups. Received July 14, 1999 / Revised May 17, 2000 / Published online February 5, 2001     

Hyperbolic spaces at large primes and a conjecture of Moore

A simply connected finite complex X is called elliptic if its rational homotopy Lie algebra is of finite dimension and hyperbolic otherwise. According to a conjecture of Moore, there exists an exponent for the p-torsion part of if and only if X is elliptic. In this note, it is shown that, provided the prime p is sufficiently large, a hyperbolic space with p-torsion free loop space homology has no exponent in the p-torsion of the homotopy groups. For a class of formal spaces, this result is obtained for every odd prime.     

Le genre de Mislin des espaces rationnellement équivalents à un produit de deux sphères de dimensions différentes

 Zabrodsky exact sequences are algebraic tools which express the genus set of a space X in term of its self-maps, when X has the rational homotopy type of a co-ℋ-space or an ℋ-space. Explicit examples show these methods can't be generalized to the class of all simply connected finite CW-complexes. We however construct a Zabrodsky exact sequence for those three cells CW-complexes rationally equivalent to the product of two spheres S ^k ×S ⁿ , n>k≥2. We deduce, from results of Morisugi-Oshima, the genus of some spherical bundles. Received: 17 March 2001 / Revised version: 8 August 2001     

Cochain model for thickenings and its application to rational LS-category

A thickening of a finite CW-complex X is by definition a compact manifold M of the same simple homotopy type as X. We give a model for the cochain complex of the boundary of that manifold, C ^*(δM), as a module over the cochain algebra C ^*(X). We also show how to construct an algebraic model of the rational homotopy type of δC ^*(M) from a model of X. Using this rational model, we prove a new formula for the rational Lusternik–Schnirelmann category of X. Received: 24 September 1999     

Separated Lie models and the homotopy Lie algebra

A simply connected topological space X has homotopy Lie algebra π_∗(ΩX)⊗Q. Following Quillen, there is a connected differential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X, and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property that we call being separated. The homology of a separated dgL has a particular form which lends itself to calculations.     

Rationalized evaluation subgroups of a map II: Quillen models and adjoint maps

We identify the long exact sequence induced on rational homotopy groups by the evaluation map , and in particular the rationalization of the evaluation subgroups of f, in terms of derivations of Quillen models and adjoint maps. We consider a generalization of a question of Gottlieb within the context of rational homotopy theory. We also study the rationalization of the G-sequence of a map. In a separate result of independent interest, we give an explicit Quillen minimal model of a product A×X, in the case in which A is a rational co-H-space.     

PHANTOM MAPS,COGROUPS AND THE SUSPENSION MAP

ABSTRACT

The set Ph(X, Y) of pointed homotopy classes of phantom maps from X to Y admits a natural group structure if either Y is a grouplike space or X is a cogroup. In the present paper, the group structure on Ph(X,Y) is examined in the second case. (The first case was examined in an earlier paper.) The results in the two cases are similar—for instance, the group structure turns out to be abelian, divisible and independent of the grouplike structure on Y or the cogroup structure on X—but the techniques used to establish the results differ substantially in the two cases.

In addition, a study of the map g*: Ph(X,Y₁) → Ph(X,Y₂) induced by a map g: Y₁ → Y₂ of grouplike spaces is initiated. A particularly interesting special case of this situation is the suspension map Ph(X, Y) → Ph(X, ΩσY) ? Ph(σX, σY) with Y a grouplike space.     

Homotopy operations and the obstructions to being anH-space

We describe an obstruction theory for a given topological spaceX to be anH-space, in terms of higher homotopy operations and show how this theory can be used to calculate such operations in certain cases. Date: September 13, 1995     

The relationship between the diagonal and the bar constructions on a bisimplicial set

The aim of this paper is to prove that the homotopy type of any bisimplicial set X is modelled by the simplicial set , the bar construction on X. We stress the interest of this result by showing two relevant theorems which now become simple instances of it; namely, the Homotopy colimit theorem of Thomason, for diagrams of small categories, and the generalized Eilenberg-Zilber theorem of Dold-Puppe for bisimplicial Abelian groups. Among other applications, we give an algebraic model for the homotopy theory of (not necessarily path-connected) spaces whose homotopy groups vanish in degree 4 and higher.     

球面稳定同伦群中的一族新元素

本文研究了球面稳定同伦群中元素的非平凡性.利用May谱序列,证明了在Adams谱序列E₂项中存在乘积元素收敛到球面稳定同伦群的一族阶为p的非零元,此非零元具有更高维数的滤子.     

The homotopy categorical crossed module of a CW-complex

Crossed modules have longstanding uses in homotopy theory and the cohomology of groups. The corresponding notion in the setting of categorical groups, that is, categorical crosses modules, allowed the development of a low-dimensional categorical group cohomology. Now, its relevance is also shown here to homotopy types by associating, to any pointed CW-complex (X,∗), a categorical crossed module that algebraically represents the homotopy 3-type of X.     

<Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> theory,L.S. category,and strong category

Relations between category and strong category are studied. The notion of a homotopy coalgebra of order r over the Ganea comonad is introduced. It is shown that cat(X) =Cat(X) holds if a finite 1-connected complex X carries such a structure with r sufficiently large.