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

@article{Dwyer1994HomotopyFM,
  title={Homotopy fixed-point methods for Lie groups and finite loop spaces},
  author={William G. Dwyer and Clarence W. Wilkerson},
  journal={Annals of Mathematics},
  year={1994},
  volume={139},
  pages={395-442}
}
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. 

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References

SHOWING 1-10 OF 21 REFERENCES

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.