Topological arguments have been used for 30 years to prove some of the most important fundamental results in distributed computing. In 1985, Fischer, Lynch, and Paterson [1] showed, using chains of uncertainties forming a connected graph, that there is no fault-tolerant message-passing protocol for the consensus task, even if only one process may crash. This powerful insight of viewing distributed computation through the lens of combinatorial topology received heightened attention after, in 1993, it was shown by Saks and Zaharoglou [2] using topological techniques that k-set agreement is not wait-free solvable. Since then, many other important results were proven using such techniques.

This outstanding book is motivated by this history and explores the connections between distributed computation and topology in detail. The authors provide an invaluable service to the research community in distributed computing by organizing and freshly presenting these very fruitful and powerful connections. The book systematically organizes material that previously was only available across a collection of conference and journal publications with inconsistent notations and terminology, and also summarizes and explains topological techniques to non-mathematicians. It can be used as a textbook for undergraduate (Parts 1 and 2) or graduate courses in distributed computing (Parts 1 to 3) or as a reference for researchers.

The book is divided into four parts and 16 chapters. The three chapters in Part 1 cover fundamentals of combinatorial topology and explain how it helps to understand distributed computation. Chapter 1 begins with an intuitive description of the approach discussed in this book. Chapter 2 describes the approach in more detail, and chapter 3 introduces the topological notation used. The four chapters in Part 2 discuss so-called colorless tasks that can be defined without considering process identifiers. Chapter 4 describes the basic combinatorial model of computation. In chapter 5, these techniques are applied to study the solvability of colorless tasks. In chapter 6, this is extended to processes with Byzantine failures. Chapter 7 then introduces reductions that allow transformation of results from one model of computation to another.

Part 3 considers general tasks. In chapter 8, the authors generalize the mathematical framework they used in the previous chapters. In chapter 9, they consider manifold tasks. Chapter 10 then studies how computation affects connectivity, and chapter 11 presents necessary and sufficient conditions for solving general tasks in various models of computation.

Part 4 finally discusses advanced topics in distributed computing by using additional notions from topology. Chapter 12 studies the renaming task, and chapter 13 uses shellability to show that several models of computation can be analyzed with the same formal tools. Chapter 14 then studies reductions and simulations for general tasks. Chapter 15 connects a certain class of tasks with the word problem for finitely presented groups. Finally, chapter 16 uses Schlegel diagrams to prove basic topological properties of the core models of computation.