**Graduate Studies in Mathematics**

Volume: 207;
2020;
312 pp;
Softcover

MSC: Primary 57;
Secondary 05; 06; 55; 58

**Print ISBN: 978-1-4704-6455-4
Product Code: GSM/207.S**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

**Electronic ISBN: 978-1-4704-6008-2
Product Code: GSM/207.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

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#### Supplemental Materials

# Organized Collapse: An Introduction to Discrete Morse Theory

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*Dmitry N. Kozlov*

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.

#### Readership

Graduate students and researchers interested in applied topology.

#### Reviews & Endorsements

The book is well-equipped with both illustrative examples, many of them drawing on combinatorics and on graph theory, and plenty of exercises gathered at the end of each chapter. In each of the four parts of the book, suggestions for further reading are included with comments guiding a reader to a targeted exploration of the literature. I expect it to find regular use as a reference myself.

-- Mikael Vejdemo-Johansson, CUNY College of Staten Island

#### Table of Contents

# Table of Contents

## Organized Collapse: An Introduction to Discrete Morse Theory

- Preamble 1616
- Preface 1818
- Part 1 . Introduction to Homology 2626
- Chapter 1. The First Steps 2828
- Chapter 2. Simplicial Homology 4040
- 2.1. Plain simplicial complexes 4040
- 2.2. The boundary operator 4444
- 2.3. Homology of a plain simplicial complex 4848
- 2.4. Geometric realization 5555
- 2.5. Finite abstract simplicial complexes and standard constructions 5858
- 2.6. Abstract simplicial complexes on finite ordered sets 6161
- 2.7. Further constructions and associated homology 6565
- 2.8. Simplicial maps 7676
- Exercises 7777

- Chapter 3. Beyond the Simplicial Setting 8282

- Part 2 . Further Aspects of Homology Theory 104104
- Part 3 . Basic Discrete Morse Theory 170170
- Chapter 9. Simplicial Collapses 172172
- Chapter 10. Organizing Collapsing Sequences 190190
- 10.1. Face poset of an abstract simplicial complex 190190
- 10.2. Acyclic matchings 194194
- 10.3. Collapsing sequences vs acyclic matchings: Theorem A 197197
- 10.4. Collapsing sequences and cones 199199
- 10.5. Standard subdivisions of collapsible complexes 201201
- 10.6. Standard chromatic subdivision 205205
- 10.7. Combinatorial collapsing sequences 209209
- Exercises 215215

- Chapter 11. Internal Collapses and Discrete Morse Theory 218218
- 11.1. Replacing the simplicial complex with a smaller cellular complex: Theorem B 218218
- 11.2. Internal collapses and attachment maps 220220
- 11.3. Attaching cells to homotopy equivalent spaces 222222
- 11.4. Organizing internal collapses 224224
- 11.5. Examples of computation 228228
- 11.6. An acyclic matching associated to a sequence of vertices 233233
- Exercises 234234

- Chapter 12. Explicit Homology Classes Associated to Critical Cells 236236
- Chapter 13. The Critical Morse Complex 250250
- Chapter 14. Implications and Variations 258258
- Suggested further reading for Part 3 270270

- Part 4 . Extensions of Discrete Morse Theory 272272
- Chapter 15. Algebraic Morse Theory 274274
- Chapter 16. Discrete Morse Theory for Posets 296296
- Chapter 17. Discrete Morse Theory for CW Complexes 306306
- Chapter 18. Discrete Morse Theory and Persistence 312312
- Suggested further reading for Part 4 320320
- Index 322322
- List of Figures 326326
- List of Tables 330330
- Bibliography 332332