Algorithmic topology and classification of 3-manifolds.

*(English)*Zbl 1048.57001
Algorithms and Computation in Mathematics 9. Berlin: Springer (ISBN 3-540-44171-9/hbk). xii, 478 p. (2003).

This book provides a comprehensive and detailed account of different topics in algorithmic 3-dimensional 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 from the peculiar point of view of special polyhedra and special spines of 3-manifolds. This choice contributes to keep the level of the exposition really elementary. So much that, in the author’s words, “elementary knowledge of topology and algebra is required, but understanding the concepts of “topological space” and “group” is quite sufficient for most of the book”. Nevertheless, in the reviewer’s opinion, some familiarity with classical topology of 3-manifolds is highly recommendable.

Here is an outline of the contents of the book. Chapter 1 is aimed at introducing simple and special polyhedra and spines, and proving the basic facts about them. In particular, moves relating different special spines of the same 3-manifolds are discussed, starting from the early results of the author [Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 5, 1104–1116 (1987; Zbl 0642.57003)] and the reviewer [Topology 3rd Natl. Meet., Trieste/Italy 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 18, 391–414 (1988; Zbl 0672.57004)]. Chapter 2 deals with the notion of complexity of a 3-manifold, defined in terms of its almost simple spines. After having presented the classification of the 3-manifolds up to complexity 6, the author shows that all the 3-manifolds with complexity \(\leq 8\) are graph manifolds and describes the first hyperbolic 3-manifold of complexity 9. Chapters 3 and 4 contain a nice exposition of the theory of Haken’s normal surfaces and of some algorithms derived from it, such as the ones for testing (boundary) incompressibility of a given surface and detecting sufficiently large 3-manifolds. Normal surfaces are considered in both the contexts of triangulations and handle decompositions (specially the ones associated to simple spines). This last approach is adopted in Chapter 5, in order to obtain the Rubinstein-Thompson algorithm for recognition of the 3-sphere. Chapter 6 is devoted to the algorithmic recognition of Haken manifolds. This rather long chapter provides an essentially self-contained proof of the recognition theorem, including some original contribution by the author. In particular, existence and uniqueness of JSJ-decompositions are proved and Stallings and quasi-Stallings manifolds are discussed in detail. Here, simple polyhedra are used to effectively encode hierarchies. Chapters 7 and 8, respectively, are concerned with computer implementation of the recognition theorem and with Turaev-Viro invariants, in order to describe an enumeration procedure for 3-manifolds of low complexity, by Martelli and Petronio. The resulting 3-manifolds up to complexity 6 are listed in Chapter 9 (Appendix), together with their minimal special spines and their Turaev-Viro invariants.

For its contents and the point of view adopted in their exposition, this book has very little overlap with other ones in the literature. In spite of the similarity in the title, this is true also for the previous book “Algorithmic and computer methods in three-dimensional topology” by the author and A. T. Fomenko [Algorithmic and computer methods in three-dimensional topology (1991; Zbl 0748.57005), Algorithmic and computer methods for three-manifolds (1997; Zbl 0885.57009) or Algorithmic and computer methods in three-dimensional topology (1998; Zbl 0906.57001)].

Concerning the style the author succeeded to conform to the ground rules stated in the preface, as for completeness, modularity and clarity of the text. Many figures integrate the exposition. Hopefully, misprints will be eliminated in a next printing.

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”.

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 from the peculiar point of view of special polyhedra and special spines of 3-manifolds. This choice contributes to keep the level of the exposition really elementary. So much that, in the author’s words, “elementary knowledge of topology and algebra is required, but understanding the concepts of “topological space” and “group” is quite sufficient for most of the book”. Nevertheless, in the reviewer’s opinion, some familiarity with classical topology of 3-manifolds is highly recommendable.

Here is an outline of the contents of the book. Chapter 1 is aimed at introducing simple and special polyhedra and spines, and proving the basic facts about them. In particular, moves relating different special spines of the same 3-manifolds are discussed, starting from the early results of the author [Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 5, 1104–1116 (1987; Zbl 0642.57003)] and the reviewer [Topology 3rd Natl. Meet., Trieste/Italy 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 18, 391–414 (1988; Zbl 0672.57004)]. Chapter 2 deals with the notion of complexity of a 3-manifold, defined in terms of its almost simple spines. After having presented the classification of the 3-manifolds up to complexity 6, the author shows that all the 3-manifolds with complexity \(\leq 8\) are graph manifolds and describes the first hyperbolic 3-manifold of complexity 9. Chapters 3 and 4 contain a nice exposition of the theory of Haken’s normal surfaces and of some algorithms derived from it, such as the ones for testing (boundary) incompressibility of a given surface and detecting sufficiently large 3-manifolds. Normal surfaces are considered in both the contexts of triangulations and handle decompositions (specially the ones associated to simple spines). This last approach is adopted in Chapter 5, in order to obtain the Rubinstein-Thompson algorithm for recognition of the 3-sphere. Chapter 6 is devoted to the algorithmic recognition of Haken manifolds. This rather long chapter provides an essentially self-contained proof of the recognition theorem, including some original contribution by the author. In particular, existence and uniqueness of JSJ-decompositions are proved and Stallings and quasi-Stallings manifolds are discussed in detail. Here, simple polyhedra are used to effectively encode hierarchies. Chapters 7 and 8, respectively, are concerned with computer implementation of the recognition theorem and with Turaev-Viro invariants, in order to describe an enumeration procedure for 3-manifolds of low complexity, by Martelli and Petronio. The resulting 3-manifolds up to complexity 6 are listed in Chapter 9 (Appendix), together with their minimal special spines and their Turaev-Viro invariants.

For its contents and the point of view adopted in their exposition, this book has very little overlap with other ones in the literature. In spite of the similarity in the title, this is true also for the previous book “Algorithmic and computer methods in three-dimensional topology” by the author and A. T. Fomenko [Algorithmic and computer methods in three-dimensional topology (1991; Zbl 0748.57005), Algorithmic and computer methods for three-manifolds (1997; Zbl 0885.57009) or Algorithmic and computer methods in three-dimensional topology (1998; Zbl 0906.57001)].

Concerning the style the author succeeded to conform to the ground rules stated in the preface, as for completeness, modularity and clarity of the text. Many figures integrate the exposition. Hopefully, misprints will be eliminated in a next printing.

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”.

Reviewer: Riccardo Piergallini (Camerino)

##### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57M99 | General low-dimensional topology |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57M50 | General geometric structures on low-dimensional manifolds |