Organized Collapse: an Introduction to Discrete Morse Theory: (Graduate Studies in Mathematics)

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.

Table of Contents:
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