Homotopy fixed-point methods for Lie groups and finite loop spaces

  title={Homotopy fixed-point methods for Lie groups and finite loop spaces},
  author={William G. Dwyer and Clarence W. Wilkerson},
  journal={Annals of Mathematics},
A loop space X is by definition a triple (X, BX, e) in which X is a space, BX is a connected pointed space, and e: X -QBX is a homotopy equivalence from X to the space QBX of based loops in BX. We will say that a loop space X is finite if the integral homology H*(X, Z) is finitely generated as a graded abelian group, i.e., if X appears at least homologically to be a finite complex. In this paper we prove the following theorem. 

Higher homotopy commutativity of H-spaces with finitely generated cohomology

Suppose X is a simply connected mod p H-space such that the mod p cohomology H * (X;Z/p) is finitely generated as an algebra. Our first result shows that if X is an An-space, then X is the total


The theory of p-local compact groups, developed in an earlier paper by the same authors, is designed to give a unified framework in which to study the p-local homotopy theory of classifying spaces of

Chevalley p–local finite groups

The concept of a p–compact group was introduced by Dwyer and Wilkerson in [25] as a p–local homotopy theoretic analogue of a compact Lie group. A p–compact group is a triple .X;BX; e/, where H .X I

A torus theorem for homotopy nilpotent loop spaces

Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces

The finiteness obstruction for loop spaces

Abstract. For finitely dominated spaces, Wall constructed a finiteness obstruction, which decides whether a space is equivalent to a finite CW-complex or not. It was conjectured that this finiteness

Product splittings for p-compact groups

The purpose of this paper is to prove a theorem similar to 1.1 for p-compact groups, which are homotopy theoretic analogues of compact Lie groups. Suppose that X is a p-compact group (see §2) with

A finite loop space not rationally equivalent to a compact Lie group

We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group,

Homotopy type and v1-periodic homotopy groups of p-compact groups


The problem of deciding whether or not a given topological space X is homotopy equivalent to a loop space is of classical interest in homotopy theory. Moreover, given that X is a loop space, one



Homotopical uniqueness of classifying spaces

The cohomology ring of a finite group

Using the classifying space of the group, Venkov [9] has provided a short and elegant proof of the finite generation of H*(G, Z) for G in a class of groups including the class of finite groups.

The homotopic uniqueness of BS3

1.2 Remark. It is easy to see that 1.1 implies that there is up to homotopy only one space B whose loop space is homotopy equivalent to the p-completion of S. To get such a strong uniqueness result

Homological dimension in local rings

Introduction. This paper is devoted primarily to the study of commutative noetherian local rings. The main task is to compare purely algebraic properties with properties of a homological nature. A

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

The localization of spectra with respect to homology

Smith theory and the functorT

J. Lannes has introduced and studied a remarkable functor T [L1] which takes an unstable module (or algebra) over the Steenrod algebra to another object of the same type. This functor has played an

On the Structure of Hopf Algebras

induced by the product M x M e M. The structure theorem of Hopf concerning such algebras has been generalized by Borel, Leray, and others. This paper gives a comprehensive treatment of Hopf algebras

Simplicial objects in algebraic topology

Since it was first published in 1967, Simplicial Objects in Algebraic Topology has been the standard reference for the theory of simplicial sets and their relationship to the homotopy theory of

The realization of polynomial algebras as cohomology rings.