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Reviewer: Burkhard Englert

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. Online Computing Reviews Service

Reviewer: Shrisha Rao

Topology is a relatively new branch of mathematics, growing and maturing since the early 20th century, though its roots can probably be traced to much earlier. Some basic concepts from topology, like topological invariance (often expressed by saying “a mug is homeomorphic to a doughnut”), are fairly well known to a large number of scientifically literate amateurs, and the subject has had a variety of applications, which are perhaps less well known. It has developed strong bonds with other branches of mathematics, including set theory, knot theory, geometry, and analysis. It is claimed by some that the impetus for using topology in distributed computing arose from Fischer et al.'s seminal paper [1], which used a topological line of reasoning for a problem space described as a hypercube to arrive at an important impossibility result. While this result itself was of great interest, it also spurred on some researchers working in distributed computing to explore further connections and apply topological arguments to prove results in distributed computing [2,3,4]. Somewhat roughly speaking, the idea is to reduce the space of some problem in distributed computing to a topological structure (a simplicial complex, or a manifold), with a solution to the problem corresponding to some valid transformation of the structure. If it is known (or can be shown) based on the theory and results of topology that such a transformation is not possible, it then follows that the distributed computing problem has no solution either. Thus, this is a newer tool for impossibility results in distributed computing, and many such results previously proved otherwise have later been recast to use topology. Until recently, there has not been a monograph that comprehensively covers the intersection of topology and distributed computing, with some books [5,6] having no coverage. This book thus finds its place for filling precisely this niche, and will be welcomed by readers who are thankful for not having to pore through the literature in hopes of attaining a modicum of comprehension. It covers its subject essentially from the first principles (though a prior understanding of the basics of distributed algorithms, and of algebraic topology, would serve the reader well). The book consists of four parts. Part 1, “Fundamentals,” covers in three chapters the bare essentials of distributed algorithms and algebraic topology (which the authors refer to by its older name, combinatorial topology, and describe as “a higher-dimensional version of graph theory,” which may or may not be a description that satisfies most mathematicians). Part 2, “Colorless Tasks,” contains four chapters and covers the basics of wait-free algorithms (called “colorless” because only the input values of the processors are important, not which processors had what values) for consensus/agreement under shared memory as well as message passing. Part 3, “General Tasks,” covers general-that is, “colored”-tasks over four chapters. Some advanced topics, mainly recent advances by the authors and others, are covered in Part 4, “Advanced Topics.” While the authors' writing is quite lucid and should invite few serious complaints from either computer scientists or mathematicians, it is probably true that relatively few computer scientists will find it easy to go through (mathematicians will, in all likelihood, fare much better, even if they know little of distributed computing to begin with). Computer science students and established researchers alike will probably find it easier to understand this work if they have taken a basic class in topology at the advanced undergraduate or basic graduate level, or at least have spent some time studying a standard textbook [7,8]. Online Computing Reviews Service