DISCRETE MORSE THEORY

Cover1
Title page2
Preface10
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 theory28
Chapter 1. Simplicial complexes30

1.1. Basics of simplicial complexes30
1.2. Simple homotopy46
Chapter 2. Discrete Morse theory56

2.1. Discrete Morse functions59
2.2. Gradient vector fields71
2.3. Random discrete Morse theory88
Chapter 3. Simplicial homology96

3.1. Linear algebra97
3.2. Betti numbers101
3.3. Invariance under collapses110
Chapter 4. Main theorems of discrete Morse theory116

4.1. Discrete Morse inequalities116
4.2. The collapse theorem126
Chapter 5. Discrete Morse theory and persistent homology132

5.1. Persistence with discrete Morse functions132
5.2. Persistent homology of discrete Morse functions149
Chapter 6. Boolean functions and evasiveness164

6.1. A Boolean function game164
6.2. Simplicial complexes are Boolean functions167
6.3. Quantifying evasiveness170
6.4. Discrete Morse theory and evasiveness173
Chapter 7. The Morse complex184

7.1. Two definitions184
7.2. Rooted forests192
7.3. The pure Morse complex194
Chapter 8. Morse homology202

8.1. Gradient vector fields revisited203
8.2. The flow complex210
8.3. Equality of homology211
8.4. Explicit formula for homology214
8.5. Computation of Betti numbers220
Chapter 9. Computations with discrete Morse theory224

9.1. Discrete Morse functions from point data224
9.2. Iterated critical complexes235
Chapter 10. Strong discrete Morse theory248

10.1. Strong homotopy248
10.2. Strong discrete Morse theory257
10.3. Simplicial Lusternik-Schnirelmann category264
Bibliography272
Notation and symbol index280
Index

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.

Author
Nicholas A. Scoville: Ursinus College, Collegeville, PA.