Uniform Classification of Open Manifolds


Series: Mathematics Research Developments
BISAC: MAT012020

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 adapted by 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 describes geometric generators is 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. In particular, guidance is given on the ways in which supply chains can be diagnosed for vulnerabilities and the remedies that may be applied. One such countermeasure, virtual dualization, is explained in detail as a means for achieving both supply chain robustness and competitiveness for complex products that require intense coordination in their design and production. A common theme that runs throughout the chapter is the importance of building trust among the participants in a supply chain. (Imprint: Nova)

Table of Contents

Table of Contents

Chapter 1. Introduction

Chapter 2. Uniform Structures of Proper Metric Spaces, Riemannian Manifolds and Vector Bundles

Chapter 3. Uniform Complexes and Their Algebraic Topology

Chapter 4. Geometric and Analytic Bordism for Open Manifolds

Chapter 5. Open Poincare Complexes

Chapter 6. Surgery to Proper Homotopy Equivalence at Infinity

Chapter 7. Surgery to Uniform Homotopy Equivalence

Chapter 8. Appendix 1: Inverse and Direct Systems

Chapter 9. Appendix 2: Wall-Novikov Groups and Lagrangian Calculus

Chapter 10. Appendix 3: N-Ads and Surgery Bordism Groups



This book was reviewed in Mathematical Reviews Clippings – November 2018

Published in the journal, Zentralblatt Math. To read the review, click here.


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