##
**Three-dimensional geometry and topology. Vol. 1. Ed. by Silvio Levy.**
*(English)*
Zbl 0873.57001

Princeton Mathematical Series. 35. Princeton, NJ: Princeton University Press. x, 311 p. (1997).

The book originates from notes of a graduate course Thurston gave at Princeton in the period 1978-1980. These notes have been widely distributed and have become the most important and influent text in low-dimensional topology and hyperbolic geometry of the last two decades. In fact they founded the new field of hyperbolic 3-manifolds, with connections to various other subjects as Kleinian groups, reflection groups, Teichmüller theory, differential geometry etc. The notes constitute the background and the basis for the important hyperbolization theorem for Haken 3-manifolds. Written in a very intuitive way, many of the beautiful ideas have been extended and formalized by various authors, sometimes using also different approaches and methods. After 1990, the notes have been thoroughly revised and extended, and the present book presents the first four chapters of this revised version.

The first chapter (“What is a manifold”) discusses polygons and surfaces, hyperbolic surfaces and some 3-manifolds as the 3-torus, the 3-sphere and elliptic 3-space, the Poincaré and the Seifert-Weber dodecahedral spaces, lens spaces and the complement of the figure-eight knot. In the second chapter (“Hyperbolic geometry and its friends”) various models of hyperbolic spaces are discussed, their isometries, trigonometric formulas, volume formulas etc. The third chapter is on “Geometric manifolds”, that is on geometric structures on manifolds, the developing map and completeness and the eight model geometries in dimension three, containing also sections on discrete groups, bundles and connections, contact structures, PL-manifolds and smoothings. The last and fourth chapter is entitled “The structure of discrete groups”, with sections on groups generated by small elements, Euclidean manifolds and crystallographic groups, Euclidean and spherical (elliptic) 3-manifolds, the ‘thick-thin” decomposition of hyperbolic manifolds, Teichmüller space and 3-manifolds modeled on fibered geometries. (The next chapter of the revised version continues with the theory of orbifolds, in particular the classification of 2-orbifolds and geometric 3-orbifolds.)

Even more than the original notes, the book is full of ideas, examples, pictures, exercises and problems, side-remarks, hints to further developments and connections with other fields. It is written very densely, and often in a more intuitive and “experimental” than formal way (but much more complete and detailed than the original notes). This makes it both fascinating and challenging to read and one of the most beautiful and original texts in topology and geometry. Hopefully also a revised version of the other parts of the original notes will appear in the near future.

The first chapter (“What is a manifold”) discusses polygons and surfaces, hyperbolic surfaces and some 3-manifolds as the 3-torus, the 3-sphere and elliptic 3-space, the Poincaré and the Seifert-Weber dodecahedral spaces, lens spaces and the complement of the figure-eight knot. In the second chapter (“Hyperbolic geometry and its friends”) various models of hyperbolic spaces are discussed, their isometries, trigonometric formulas, volume formulas etc. The third chapter is on “Geometric manifolds”, that is on geometric structures on manifolds, the developing map and completeness and the eight model geometries in dimension three, containing also sections on discrete groups, bundles and connections, contact structures, PL-manifolds and smoothings. The last and fourth chapter is entitled “The structure of discrete groups”, with sections on groups generated by small elements, Euclidean manifolds and crystallographic groups, Euclidean and spherical (elliptic) 3-manifolds, the ‘thick-thin” decomposition of hyperbolic manifolds, Teichmüller space and 3-manifolds modeled on fibered geometries. (The next chapter of the revised version continues with the theory of orbifolds, in particular the classification of 2-orbifolds and geometric 3-orbifolds.)

Even more than the original notes, the book is full of ideas, examples, pictures, exercises and problems, side-remarks, hints to further developments and connections with other fields. It is written very densely, and often in a more intuitive and “experimental” than formal way (but much more complete and detailed than the original notes). This makes it both fascinating and challenging to read and one of the most beautiful and original texts in topology and geometry. Hopefully also a revised version of the other parts of the original notes will appear in the near future.

Reviewer: B.Zimmermann (Trieste)

### MathOverflow Questions:

Classification of surfaces and the TOP, DIFF and PL categories for manifoldsExistence of a geometric structure on a solid torus

Find a surface or 3-manifold whose fundamental group is \((\mathbb{Z}/n\mathbb{Z}) \rtimes (\mathbb{Z}/2\mathbb{Z})\)

### MSC:

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

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