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On Nielsen realization and manifold models for classifying spaces

James F. Davis, Wolfgang Lueck

Abstract

We consider the problem of whether, for a given virtually torsionfree discrete group $Γ$, there exists a cocompact proper topological $Γ$-manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. This problem is related to Nielsen Realization. We will make the assumption that the expected manifold model has a zero-dimensional singular set. Then we solve the problem in the case, for instance, that $Γ$ contains a normal torsionfree subgroup $π$ such that $π$ is hyperbolic and $π$ is the fundamental group of an aspherical closed manifold of dimension greater or equal to five and $Γ/π$ is a finite cyclic group of odd order.

On Nielsen realization and manifold models for classifying spaces

Abstract

We consider the problem of whether, for a given virtually torsionfree discrete group , there exists a cocompact proper topological -manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. This problem is related to Nielsen Realization. We will make the assumption that the expected manifold model has a zero-dimensional singular set. Then we solve the problem in the case, for instance, that contains a normal torsionfree subgroup such that is hyperbolic and is the fundamental group of an aspherical closed manifold of dimension greater or equal to five and is a finite cyclic group of odd order.
Paper Structure (31 sections, 17 theorems, 121 equations)

This paper contains 31 sections, 17 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. A special case
  3. Slice manifold systems and slice manifold models
  4. Main theorems
  5. Acknowledgments
  6. Relating some conditions on $\Gamma$
  7. Equivariant homology theories, spectra over groupoids and the Full Farrell-Jones Conjecture
  8. Equivariant homology theories
  9. Equivariant homology theories from spectra over groupoids
  10. Some basics about $K$-and $L$-theory of groups rings
  11. The Full Farrell-Jones Conjecture
  12. Computing the $K$-theory
  13. The definition of Whitehead groups
  14. Computing Whitehead groups
  15. Computing the $L$-theory
  16. ...and 16 more sections

Key Result

Theorem 1.3

Suppose there is a short exact sequence of groups with $G$ finite. Suppose that the following conditions are satisfied: Then:

Theorems & Definitions (48)

  • Definition 1.2: Conditions on $\Gamma$
  • Theorem 1.3: Oriented manifold models
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.10: Slice manifold model
  • Definition 1.11: Condition (H)
  • Definition 1.12: Condition (S)
  • ...and 38 more