The geometries of 3-manifolds.

*(English)*Zbl 0561.57001In 1978/79 Thurston formulated his now famous ”geometrization-conjecture” claiming that every 3-manifold is decomposable in a natural way into geometric pieces. Thus emphazising the rôle of geometry in the theory of 3-manifolds. Classical work of Hopf, Seifert-Threlfall, Milnor etc. and recent work of Thurston itself gives evidence for this conjecture. If true this conjecture would not only provide a very unifying picture for 3-manifolds in the whole (to some extend this would also be supplied by the conjecture that every irreducible 3-manifold is either a Seifert fibre space or a virtually Haken 3-manifold), but would at the same time also offer new and powerful tools for dealing with 3-manifolds in general. Thurston’s conjecture is, however, far reaching and includes e.g. the long-standing Poincaré- and generalized Smith-conjecture.

In § 6 of the paper under review the status of the geometrization- conjecture is briefly discussed. Moreover, this paper is concerned with the existence and uniqueness of geometric structures for 3-manifolds, leaving almost untouched the hyperbolic structures (a first good introduction to the latter is provided by J. W. Morgan’s article [Chapter V in ”The Smith-conjecture”, ed. by H. Bass and J. W. Morgan (1984)]). It is surveyed what is known in this direction. In § 5 the author sketches the general theory of these 3-dimensional geometric structures, first discussing what should be the appropriate definition of ”geometric structures”, and then outlining Thurston’s proof that there are only eight geometries which ought to be considered in dimension three - these being \({\mathbb{H}}^ 3\), \(S^ 3\), \(E^ 3\), \({\mathbb{H}}^ 2\times {\mathbb{R}}\), \(S^ 2\times {\mathbb{R}}\), \(_ 2{\mathbb{R}}^{\sim}\), Sol, Nil. Moreover, in theorem 5.3. (the main result) those closed 3-manifolds are classified which admit non-hyperbolic structures. It turns out, in particular, that all of them are Seifert fibre spaces or torus (Klein bottle)-bundles over \(S^ 1\). Thus the paper can also be considered as essentially being an introduction to Seifert fibre spaces from the geometric point of view. However, to prove theorem 5.3. needs preparation, in particular the basic topological theory of Seifert fibre spaces, and the paper provides this theory in §§ 1 and 3, using the language of orbifolds throughout its exposition. In § 2 2-dimensional orbifolds and the definitions of covering and Euler number of orbifolds are discussed.

The main part of the paper, however, is § 4. Here the eight 3- dimensional geometries are carefully introduced. In each case (except the hyperbolic one) the discrete groups of isometries which act freely with compact quotients are classified, exhibiting, in the spirit of Seifert, the existence of invariant foliations. This survey is lucidly written and well-motivated, so that everyone interested in 3-manifolds will enjoy reading it.

In § 6 of the paper under review the status of the geometrization- conjecture is briefly discussed. Moreover, this paper is concerned with the existence and uniqueness of geometric structures for 3-manifolds, leaving almost untouched the hyperbolic structures (a first good introduction to the latter is provided by J. W. Morgan’s article [Chapter V in ”The Smith-conjecture”, ed. by H. Bass and J. W. Morgan (1984)]). It is surveyed what is known in this direction. In § 5 the author sketches the general theory of these 3-dimensional geometric structures, first discussing what should be the appropriate definition of ”geometric structures”, and then outlining Thurston’s proof that there are only eight geometries which ought to be considered in dimension three - these being \({\mathbb{H}}^ 3\), \(S^ 3\), \(E^ 3\), \({\mathbb{H}}^ 2\times {\mathbb{R}}\), \(S^ 2\times {\mathbb{R}}\), \(_ 2{\mathbb{R}}^{\sim}\), Sol, Nil. Moreover, in theorem 5.3. (the main result) those closed 3-manifolds are classified which admit non-hyperbolic structures. It turns out, in particular, that all of them are Seifert fibre spaces or torus (Klein bottle)-bundles over \(S^ 1\). Thus the paper can also be considered as essentially being an introduction to Seifert fibre spaces from the geometric point of view. However, to prove theorem 5.3. needs preparation, in particular the basic topological theory of Seifert fibre spaces, and the paper provides this theory in §§ 1 and 3, using the language of orbifolds throughout its exposition. In § 2 2-dimensional orbifolds and the definitions of covering and Euler number of orbifolds are discussed.

The main part of the paper, however, is § 4. Here the eight 3- dimensional geometries are carefully introduced. In each case (except the hyperbolic one) the discrete groups of isometries which act freely with compact quotients are classified, exhibiting, in the spirit of Seifert, the existence of invariant foliations. This survey is lucidly written and well-motivated, so that everyone interested in 3-manifolds will enjoy reading it.

Reviewer: K.Johannson

##### MSC:

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

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

51M10 | Hyperbolic and elliptic geometries (general) and generalizations |

51M05 | Euclidean geometries (general) and generalizations |

##### MathOverflow Questions:

Existence of a geometric structure on a solid torusHomology sphere with $\mathbb{R}^3$ as the universal cover

Implications of Geometrization conjecture for fundamental group