This self-contained book by a leading topologist is devoted to algorithmic low-dimensional topology, a branch of mathematics that has recently been undergoing an intense development. The book contains plenty of important fundamental material, which is carefully presented, and also some of the author's own original contributions. For the first time ever, it gives a full exposition of the complexity theory of 3-manifolds and a complete proof of the solution of the homeomorphism problem for Haken manifolds. Written in a clear and succinct manner the book adopts a lively style, with a lot of useful pictures, making it enjoyable for those who like visual topology. The proofs are careful and detailed. This book fills a gap in the existing literature and will become a standard reference for algorithmic 3-dimensional topology both for graduate students and researchers.
From the reviews:
"This book provides a comprehensive and detailed account of different topics in algorithmic 3-dimensiona] topology, culminating with the recognition procedure for Haken manifolds and including the up-to-date results in Computer enumeration of 3-manifolds. Originating from lecture notes of various courses given by the author over a decade, the book is intended to combine the pedagogical approach of a graduate textbook (without exercises) with the completeness and reliability of a research monograph. This motivates the about 400 pages length of the main text.
All the material, with few exceptions, is presented froni the peculiar point of view of special polyhedra and special spines of 3-manifolds. This choice contributes to keep the level of the exposilion really elementary. ???.
In conclusion, the reviewer subscribes to the quotation from the back cover: "the book fills a gap in the existing literature and will become a standard reference for algorithmic 3-dimensional topology both for graduate students and researchers."
Riccardo Piergallini (Camerino), Zentralblatt f??r Mathematik 1048 (2004)
"The purpose of the book is to present a detailed overview of the algorithmis aspects of 3-manifold topology. As noted by the author, the book is largely self-contained, though basic topology and group theory are assumed. The book contains extensive references, as the author makes use of many statemants from primary sources, and a large number of figures that help the reader follow the exposition. On the whole, the book is well organized. ..."
James W. Anderson, Mathematical Reviews, Clippings from Issue 2004i
"Almost coincident with the 1985 publication ofthe first edition of Gerard Burde and Heiner Zieschang's otherwise magisterial Knots, V. Jones's revolutionary "polynomial invariants" blew that subject wide open. History now repeats itself. ... Just as Matveev ... provides a state-of-the-art fast track into all the mathematics surrounding the notorious Poincar?? ... conjecture, G. Perelman appears to have proved not only the Poincar?? conjecture, but the whole Thurston geometrization conjecture, holy grail of three-dimensional topology. ... Summing up: Recommended. Upper-division undergraduates through faculty."
D.V. Feldman, Choice - Current Reviews for College Libraries 2003
"This book provides a comprehensive and detailed account of different topics in algorithmic 3-dimensional topology ??? including the up-to-date results in computer enumeration of 3-manifolds. ??? Concerning the style the author succeeded to conform to the ground rules ??? as for completeness, modularity and clarity of the text. ??? the reviewer subscribes to the quotation from the back cover: ???the book fills a gap in the existing literature and will become a standard reference for algorithmic 3-dimensional topology both for graduate students and researchers.???"
(Riccardo Piergallini, Zentralblatt MATH, Vol. 1048, 2004)
"The purpose of this book is to present a detailed overview of the algorithmic aspects of 3-manifold topology. ??? The book contains extensive references, as the author makes use of many statements from primary sources and a large number of figures that help the reader follow the exposition. On the whole, the book is well organized."
(James W. Anderson, Mathematical Reviews, 2004 i)
