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Discrete Morse Theory

Nicholas A. Scoville
Publication Date: 
Number of Pages: 
Student Mathematical Library
[Reviewed by
Matthew Wright
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Discrete Morse Theory is the study of discrete topological spaces from the perspective of certain functions defined on such spaces. Via discrete Morse functions and discrete vector fields on these spaces, discrete Morse theory can provide homotopy-invariant simplifications, aid in the computation of homology, and more. The subject finds applications within mathematics, and to computer science and other fields. In recent years, it has also appeared prominently in topological data analysis.
In this book, Nicholas Scoville provides an extremely accessible introduction to discrete Morse theory, aimed at an undergraduate audience. The minimal mathematical background required for this text sets it apart from more advanced texts on discrete Morse theory (such as Kevin Knudson’s Morse Theory: Smooth and Discrete or Dmitry Kozlov’s recent text, Organized Collapse: An Introduction to Discrete Morse Theory). Scoville assumes only that the reader has the ability to read and write mathematical proofs. The theory of simplicial complexes is developed from the ground up. Necessary concepts from topology and combinatorics are introduced and explained as needed. A prior course in linear algebra is helpful, but not required, as Scoville explains the linear algebra necessary to understand simplicial homology in Chapter 3. To keep things accessible, he focuses on unreduced homology over the two-element field, which he defines using rank and nullity instead of quotient spaces. Quotient spaces appear later in Chapter 8, and even then only covertly as equivalence classes.
Scoville’s exposition is easy to read, with friendly discussion of technical results and plenty of examples. The text focuses on simplicial complexes, which keeps things accessible to students who have not studied topology. Scoville mentions the more general CW complexes in only a few places, pointing the reader to other sources for details. Frequent diagrams make the simplicial complexes, posets, and discrete vector fields feel quite tangible. Exercises that prompt the reader to perform computations and supply proofs are interspersed throughout the text, rather than being collected at the end of each chapter.
Chapters 1 to 3 set the stage for the discrete Morse inequalities in Chapter 4. These inequalities, the main theorems of the text, relate the Betti numbers to the numbers of critical simplices of a simplicial complex. Scoville introduces persistent homology in Chapter 5, which connects discrete Morse theory to topological data analysis. Chapter 8 further develops homological ideas, presenting Morse homology and its equivalence to simplicial homology. Chapters 6 and 9 present applications to computer science: boolean functions and evasiveness in Chapter 6, and algorithms for discrete Morse computations on simplicial complexes in Chapter 9. The remaining two chapters are more mathematically advanced; Chapter 7 studies the Morse complex, and Chapter 10 introduces strong discrete Morse theory, with connections to various other topological concepts. 
This text is well-suited for an undergraduate course, a directed study, or as a supplemental text in a course on topology or combinatorics. For readers who want to explore a particular topic in greater depth, Scoville includes 158 references to other literature. Overall, this text is an accessible, engaging introduction to a beautiful and useful theory that is ready to play a greater role in undergraduate mathematics.


Matthew Wright teaches at St. Olaf College and works in applied and computational topology. Contact him at, or visit his web site,