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Organized Collapse: An Introduction to Discrete Morse Theory

Dmitry N. Kozlov
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics
[Reviewed by
Mikael Vejdemo-Johansson
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Discrete Morse Theory is the study of discrete spaces (simplicial complexes, polyhedral complexes, CW complexes) through the use of a Morse function defined on the space. In the conceptual setup of classical Morse theory, gradient flows on a manifold equipped with a differentiable function allows the simplification of the manifold to a CW-complex determined by critical points of the Morse function itself. Robin Forman originally developed discrete Morse Theory through the study of a discrete Morse function on a cell complex, determined by conditions on the ordering of function values on faces and cofaces. Such a function partitions the complex into paired-up cells – corresponding to allowable elementary collapses – and critical (un-paired) cells. The connection between this discrete Morse pairing and elementary collapses means that the critical cells taken in isolation describe a complex that is homotopy equivalent to the starting space, thus (sometimes dramatically) simplifying the structure of the discrete space.
After many years of introductions being primarily available through articles, we have in recent years seen the release of several textbooks on the topic: first out was Kevin Knudson’s Morse Theory: Smooth and Discrete, which packages both classical (smooth) and discrete Morse theory into one volume. Within a short period of time, we then saw the release of Nicholas Scoville’s Discrete Morse Theory as well as Dmitry Kozlov’s Organized Collapse: An Introduction to Discrete Morse Theory. These two almost simultaneous books nevertheless fill quite different niches for delving into Discrete Morse Theory as a topic.
With Kozlov’s previous book on Combinatorial Algebraic Topology in mind, I was expecting a thorough treatment, well-anchored in combinatorial applications as well as applications in algebra, topology, and more – and Kozlov delivers on these expectations. The book goes much deeper into combinatorics and even algebra than Scoville's book, but also expects more from the reader. Kozlov writes out a concrete expectation of familiarity with linear algebra, group theory, and point-set topology. Critically important from these topics are knowing quotient groups and the classification of finitely generated abelian groups, and having familiarity with vector spaces over the field with 2 elements. Additionally, parts of the book expect rings, modules, tensor products, and categories to be familiar ideas. For some of the applications, both combinatorics and graph theory are quite beneficial areas to know.
For the investment – which should be comfortably accessible for an advanced undergraduate mathematics major, and certainly within reach for graduate students and active researchers – the reader is rewarded first with a thorough introduction to the homology and homotopy theory of finite and infinite simplicial complexes, to the algebra of chain complexes, and how the developed theory generalizes to cell complexes and to singular homology. I had reason several times over to reach for this section myself in research discussions with my graduate students while reading the book.
Next, Kozlov continues with a thorough treatment of discrete Morse theory itself, starting out with characterizing and developing it as a book-keeping tool for keeping track of elementary collapses and of homotopy equivalences in simplicial complexes. Once the ideas are firmly in place, Kozlov goes on to generalize in several different directions: discrete Morse theory for cellular complexes, for algebraic chain complexes, and even for poset maps with small fibers.  As a final flourish, the book ends with a quick overview of persistent homology, and in particular states exact compatibility conditions to define discrete Morse functions that work with persistent homology to produce the same kind of simplifications that Morse theory delivers for a persistent setting. While this section leaves me with more questions than answers (such as “How efficiently can this be done?” and “How restrictive are the compatibility conditions in a practical setting?”), it is an important seed of what I expect to be a fruitful direction of research in the field of persistent homology.
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 teaches at CUNY College of Staten Island and at the CUNY Graduate Center, and has worked in computational homological algebra and in applied and computational topology, in particular topological data analysis. He maintains a research community website for topological data analysis at