Organized collapse : an introduction to discrete Morse theory
著者
書誌事項
Organized collapse : an introduction to discrete Morse theory
（Graduate studies in mathematics, 207）
American Mathematical Society, c2020
- : pbk
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注記
Includes bibliographical references (p. 307-312) and index
内容説明・目次
内容説明
Applied topology is a modern subject which emerged in recent years at a crossroads of many methods, all of them topological in nature, which were used in a wide variety of applications in classical mathematics and beyond. Within applied topology, discrete Morse theory came into light as one of the main tools to understand cell complexes arising in different contexts, as well as to reduce the complexity of homology calculations.
The present book provides a gentle introduction into this beautiful theory. Using a combinatorial approach-the author emphasizes acyclic matchings as the central object of study. The first two parts of the book can be used as a stand-alone introduction to homology, the last two parts delve into the core of discrete Morse theory. The presentation is broad, ranging from abstract topics, such as formulation of the entire theory using poset maps with small fibers, to heavily computational aspects, providing, for example, a specific algorithm of finding an explicit homology basis starting from an acyclic matching.
The book will be appreciated by graduate students in applied topology, students and specialists in computer science and engineering, as well as research mathematicians interested in learning about the subject and applying it in context of their fields.
目次
Preamble
Preface
The idea of homology
The idea of discrete Morse theory
A sample application
How to use this book
Prerequisites
Guide to the literature
Part 1 . Introduction to Homology
Chapter 1. The First Steps
Chapter 2. Simplicial Homology
Chapter 3. Beyond the Simplicial Setting
Part 2 . Further Aspects of Homology Theory
Chapter 4. Category of Chain Complexes
Chapter 5. Chain Homotopy
Chapter 6. Connecting Homomorphism
Chapter 7. Singular Homology
Chapter 8. Cellular Homology
Suggested further reading for Parts 1 and 2
Part 3 . Basic Discrete Morse Theory
Chapter 9. Simplicial Collapses
Chapter 10. Organizing Collapsing Sequences
Chapter 11. Internal Collapses and Discrete Morse Theory
Chapter 12. Explicit Homology Classes Associated to Critical Cells
Chapter 13. The Critical Morse Complex
Chapter 14. Implications and Variations
Suggested further reading for Part 3
Part 4 . Extensions of Discrete Morse Theory
Chapter 15. Algebraic Morse Theory
Chapter 16. Discrete Morse Theory for Posets
Chapter 17. Discrete Morse Theory for CW Complexes
Chapter 18. Discrete Morse Theory and Persistence
Suggested further reading for Part 4
Index
List of Figures
List of Tables
Bibliography
Index
Preview Material
Preface
Table of Contents
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