Discrete Morse Theory (Student Mathematical Library, 90)

Discrete Morse theory is a powerful tool combining ideas in both topology and combinatorics. Invented by Robin Forman in the mid 1990s, discrete Morse theory is a combinatorial analogue of Marston Morse's classical Morse theory. Its applications are vast, including applications to topological data analysis, combinatorics, and computer science. This book, the first one devoted solely to discrete Morse theory, serves as an introduction to the subject. Since the book restricts the study of discrete Morse theory to abstract simplicial complexes, a course in mathematical proof writing is the only prerequisite needed. Topics covered include simplicial complexes, simple homotopy, collapsibility, gradient vector fields, Hasse diagrams, simplicial homology, persistent homology, discrete Morse inequalities, the Morse complex, discrete Morse homology, and strong discrete Morse functions. Students of computer science will also find the book beneficial as it includes topics such as Boolean functions, evasiveness, and has a chapter devoted to some computational aspects of discrete Morse theory. The book is appropriate for a course in discrete Morse theory, a supplemental text to a course in algebraic topology or topological combinatorics, or an independent study. Contents 6 Preface 10 Chapter 0. What is discrete Morse theory? 16 0.1. What is discrete topology? 17 0.2. What is Morse theory? 24 0.3. Simplifying with discrete Morse theory 28 Chapter 1. Simplicial complexes 30 1.1. Basics of simplicial complexes 30 1.2. Simple homotopy 46 Chapter 2. Discrete Morse theory 56 2.1. Discrete Morse functions 59 2.2. Gradient vector fields 71 2.3. Random discrete Morse theory 88 Chapter 3. Simplicial homology 96 3.1. Linear algebra 97 3.2. Betti numbers 101 3.3. Invariance under collapses 110 Chapter 4. Main theorems of discrete Morse theory 116 4.1. Discrete Morse inequalities 116 4.2. The collapse theorem 126 Chapter 5. Discrete Morse theory and persistent homology 132 5.1. Persistence with discrete Morse functions 132 5.2. Persistent homology of discrete Morse functions 149 Chapter 6. Boolean functions and evasiveness 164 6.1. A Boolean function game 164 6.2. Simplicial complexes are Boolean functions 167 6.3. Quantifying evasiveness 170 6.4. Discrete Morse theory and evasiveness 173 Chapter 7. The Morse complex 184 7.1. Two definitions 184 7.2. Rooted forests 192 7.3. The pure Morse complex 194 Chapter 8. Morse homology 202 8.1. Gradient vector fields revisited 203 8.2. The flow complex 210 8.3. Equality of homology 211 8.4. Explicit formula for homology 214 8.5. Computation of Betti numbers 220 Chapter 9. Computations with discrete Morse theory 224 9.1. Discrete Morse functions from point data 224 9.2. Iterated critical complexes 235 Chapter 10. Strong discrete Morse theory 248 10.1. Strong homotopy 248 10.2. Strong discrete Morse theory 257 10.3. Simplicial Lusternik-Schnirelmann category 264 Bibliography 272 Notation and symbol index 280 Index 282 Discrete,Morse,Theory

Discrete Morse theory is a powerful tool combining ideas in both topology and combinatorics. Invented by Robin Forman in the mid 1990s, discrete Morse theory is a combinatorial analogue of Marston Morse's classical Morse theory. Its applications are vast, including applications to topological data analysis, combinatorics, and computer science.This book, the first one devoted solely to discrete Morse theory, serves as an introduction to the subject. Since the book restricts the study of discrete Morse theory to abstract simplicial complexes, a course in mathematical proof writing is the only prerequisite needed. Topics covered include simplicial complexes, simple homotopy, collapsibility, gradient vector fields, Hasse diagrams, simplicial homology, persistent homology, discrete Morse inequalities, the Morse complex, discrete Morse homology, and strong discrete Morse functions. Students of computer science will also find the book beneficial as it includes topics such as Boolean functions, evasiveness, and has a chapter devoted to some computational aspects of discrete Morse theory. The book is appropriate for a course in discrete Morse theory, a supplemental text to a course in algebraic topology or topological combinatorics, or an independent study.