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Uniform Classification of Open Manifolds
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Authors: Juergen Eichhorn (Institut fuer Mathematik und Informatik, Universitaet Greifswald, Greifswald, Germany) 
Book Description:
A certain classification of open manifolds has been a continuous problem in geometry, topology and global analysis. The background for this is the fact that in any dimension greater than one, there are innumerous homotopy types. A solution to this problem seemed almost hopeless. In this monograph, the author offers an approach by decomposing the classification into two fundamental steps. This book introduces certain uniform structures in the set of all proper metric spaces under consideration (complexes, Riemannian manifolds, etc.), then considers the components of this uniform space by classifying the components by adapted homology theories (Gromov-Hausdorff and Lipschitz cohomology). The author then moves on to classifying the elements inside a component. For the second step, a geometric bordism theory for open manifolds and geometric generators are introduced. The main achievements include the invariance of a spectral gap under uniform homotopy equivalences and the vanishing of the K-theoretic signature in the geometric bordism theory. Concerning surgery, the absolutely fundamental achievements of Maumary and Taylor are included. Repeated motivations and explanations should make this monograph fairly legible to a vast audience. (Nova)


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Table of Contents:
Preface

Acknowledgements

Chapter 1. Introduction (pp. 1-8)

Chapter 2. Uniform Structures of Proper Metric Spaces, Riemannian Manifolds and Vector Bundles (pp. 9-56)

Chapter 3. Uniform Complexes and Their Algebraic Topology (pp. 57-114)

Chapter 4. Geometric and Analytic Bordism for Open Manifolds (pp. 115-178)

Chapter 5. Open Poincare Complexes (pp. 179-224)

Chapter 6. Surgery to Proper Homotopy Equivalence at Infinity (pp. 225-250)

Chapter 7. Surgery to Uniform Homotopy Equivalence (pp. 251-264)

Chapter 8. Appendix 1: Inverse and Direct Systems (pp. 265-268)

Chapter 9. Appendix 2: Wall-Novikov Groups and Lagrangian Calculus (pp. 269-270)

Chapter 10. Appendix 3: N-Ads and Surgery Bordism Groups (pp. 271-272)

References (pp. 273-278)

About the Author (pp. 279-280)

Index of Symbols(pp. 281-282)

Index of Terms (pp. 283)

Total Pages: 283

   Series:
      Mathematics Research Developments
   Binding: Hardcover
   Pub. Date: 2018 - February
   ISBN: 978-1-53612-939-7
   Status: AV
  
Status Code Description
AN Announcing
FM Formatting
PP Page Proofs
FP Final Production
EP Editorial Production
PR At Prepress
AP At Press
AV Available
  
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Uniform Classification of Open Manifolds