Organized Collapse: An Introduction to Discrete Morse Theory
About this Title
Dmitry N. Kozlov, Okinawa Institute of Science and Technology, Okinawa, Japan
Publication: Graduate Studies in Mathematics
Publication Year: 2020; Volume 207
ISBNs: 978-1-4704-5701-3 (print); 978-1-4704-6008-2 (online)
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.
Graduate students and researchers interested in applied topology.
Table of Contents
Introduction to homology
Further aspects of homology theory
- Category of chain complexes
- Chain homotopy
- Connecting homomorphism
- Singular homology
- Cellular homology
- Suggested further reading for parts 1 and 2
Basic discrete Morse theory
- Simplicial collapses
- Organizing collapsing sequences
- Internal collapses and discrete Morse theory
- Explicit homology classes associated to critical cells
- The critical Morse complex
- Implications and variations
- Suggested further reading for part 3
Extensions of discrete Morse theory
- Algebraic Morse theory
- Discrete Morse theory for posets
- Discrete Morse theory for CW complexes
- Disctrete Morse theory and persistence
- Suggested further reading for part 4