##
**Topology and combinatorics of 3-manifolds.**
*(English)*
Zbl 0820.57001

Lecture Notes in Mathematics. 1599. Berlin: Springer-Verlag. xviii, 446 p. (1995).

This work is a profound study of Heegaard splittings of compact 3- manifolds, or more specifically of Heegaard surfaces and their interactions with incompressible surfaces, in the spirit and motivated by Haken’s fundamental analysis of surfaces in 3-manifolds. In fact the main results can be considered a far-reaching generalization of Haken’s theory. The work, of 450 pages, is impressive both for the depth and generality of its results and methods as for the persistency and long breath necessary over a long period to bring it to a conclusion.

The main subject is the fundamental question of finiteness, or rigidity up to finitely many choices, of combinatorial structures on compact 3- manifolds. A combinatorial structure can be given by a triangulation or, more generally, by a Heegaard graph (for example the 1-skeleton of a triangulation) giving a Heegaard surface and Heegaard splitting of the 3- manifold into two handlebodies (possibly with boundary). Now the main question translates into the finiteness of the number of (isotopy classes of) Heegaard surfaces of a given genus in a compact 3-manifold. Note that, by classical theorems of 3-dimensional topology, such combinatorial structures exist and are unique up to stable equivalence (subdivision, stabilization), but by now many examples and classes of 3-manifolds are known with non-equivalent (minimal) Heegaard splittings. The main result of the present work says that, for a Haken-3-manifold which contains no essential Stallings fibrations (fibrations over the circle), the set of isotopy classes of Heegaard surfaces is, in fact, finite and constructible (“rigidity theorem”); this applies, for example, to all torus-free 3-manifolds with non-empty boundary.

A Haken-3-manifold is a compact irreducible 3-manifold which contains an essential (incompressible) surface. Haken proved the corresponding result for essential surfaces: a simple 3-manifold contains only a finite number of isotopy classes of such surfaces of a given Euler characteristic, and Waldhausen asked if a similar result holds for Heegard surfaces. Of course Heegaard surfaces are just on the opposite side in being totally compressible, but the present work shows – and in fact this is one of the central points – that there is a nice and fruitful interplay between these two types of surfaces. As a basic example, Haken showed that the number of intersections of an essential 2-sphere with a Heegaard surface of a 3-manifold can be reduced to one, thereby proving the additivity of the Heegaard genus under connected sums. This is generalized in the present work to the following “General handle-addition theorem” which is the starting point for the proof of the rigidity theorem: Given a compact irreducible 3-manifold, the number of intersections of essential surfaces with Heegaard surfaces is bounded above by a polynomial function of the Euler characteristics of the surfaces, after a modification of the essential surface by a finite product of Dehn twists along essential tori.

The approach to this theorem uses the notion of an \(n\)-relator 3-manifold which is a handlebody with \(n\) 2-handles attached along a system of simple closed curves on its boundary (note that every compact 3-manifold admits such a structure, modulo a completion by 3-balls or 3-handles). It is in the context of \(n\)-relator manifolds that Haken’s theory of normal surfaces in 3-manifolds is generalized. Haken’s theory was based on a fixed chosen handlebody structure of the underlying 3-manifold. In contrast, the present approach is more flexible in allowing changes of the \(n\)-relator structure by slides of the 2-handles (which do not change the homeomorphism type of the underlying 3-manifold). It is shown that every incompressible surface can be isotoped to a surface which is in “strictly normal form” with respect to a set of \(n\)-relator structures, and that the intersection of a strictly normal surface with the boundary of the handlebody from the \(n\)-relator structure can be estimated. This leads naturally to the study of surfaces in handlebodies not intersecting a given system of (attaching) curves (“relative handlebodies”).

The second chapter is devoted to the study of relative handlebodies, after the first one on handlebodies. The book then proceeds, in an organic way of increasing complexity, with the study of the basic and interesting case of 1-relator manifolds, goes on to \(n\)-relator manifolds, and finally comes to “The space of Heegaard graphs” which is the title of the concluding fifth chapter.

Changing the point of view in the above general handle-addition theorem, it can be viewed also as a theorem about Heegaard surfaces (instead of essential surfaces). Then it says that any Heegaard surface can meet a given essential surface in some limited way only. However, “To prove the rigidity theorem concerning Heegaard surfaces it is certainly not enough to consider the intersection of Heegaard surfaces with a sinlge incompressible surface alone; we rather have to study the intersections of Heegaard surfaces, or Heegaard graphs, with a complete hierarchy of surfaces”. This leads, in the final chapter, to the study of the intersections of Heegaard graphs, or more generally, Heegaard 2- complexes, with “Haken 2-complexes” which are associated to hierarchies, culminating in the above rigidity theorem for Heegaard surfaces.

The present is only a very rough outline of the contents of the book. For more details, and in particular for the many (known and new) results and applications proved for various special settings in the course of the development, we refer the reader to the introduction of the book and also to the announcement of the main results in Bull. Am. Math. Soc., New Ser. 23, 91-98 (1990; Zbl 0715.57006)].

The book is quite demanding for the reader. It offers no easy access or shortcuts to its results (the main theorem is on page 420); one has to start more or less from the beginning to get used to its mainly geometric-combinatorial, sometimes quite technical methods, with a strong emphasis on constructive and algorithmic aspects (but as so often “the way is the goal”). However, these remarks apply equally well to Haken’s fundamental work which then turned out to be at the basis of some of the most important theorems in 3-dimensional topology (such as topological rigidity and existence of hyperbolic structures for Haken-3-manifolds).

The main subject is the fundamental question of finiteness, or rigidity up to finitely many choices, of combinatorial structures on compact 3- manifolds. A combinatorial structure can be given by a triangulation or, more generally, by a Heegaard graph (for example the 1-skeleton of a triangulation) giving a Heegaard surface and Heegaard splitting of the 3- manifold into two handlebodies (possibly with boundary). Now the main question translates into the finiteness of the number of (isotopy classes of) Heegaard surfaces of a given genus in a compact 3-manifold. Note that, by classical theorems of 3-dimensional topology, such combinatorial structures exist and are unique up to stable equivalence (subdivision, stabilization), but by now many examples and classes of 3-manifolds are known with non-equivalent (minimal) Heegaard splittings. The main result of the present work says that, for a Haken-3-manifold which contains no essential Stallings fibrations (fibrations over the circle), the set of isotopy classes of Heegaard surfaces is, in fact, finite and constructible (“rigidity theorem”); this applies, for example, to all torus-free 3-manifolds with non-empty boundary.

A Haken-3-manifold is a compact irreducible 3-manifold which contains an essential (incompressible) surface. Haken proved the corresponding result for essential surfaces: a simple 3-manifold contains only a finite number of isotopy classes of such surfaces of a given Euler characteristic, and Waldhausen asked if a similar result holds for Heegard surfaces. Of course Heegaard surfaces are just on the opposite side in being totally compressible, but the present work shows – and in fact this is one of the central points – that there is a nice and fruitful interplay between these two types of surfaces. As a basic example, Haken showed that the number of intersections of an essential 2-sphere with a Heegaard surface of a 3-manifold can be reduced to one, thereby proving the additivity of the Heegaard genus under connected sums. This is generalized in the present work to the following “General handle-addition theorem” which is the starting point for the proof of the rigidity theorem: Given a compact irreducible 3-manifold, the number of intersections of essential surfaces with Heegaard surfaces is bounded above by a polynomial function of the Euler characteristics of the surfaces, after a modification of the essential surface by a finite product of Dehn twists along essential tori.

The approach to this theorem uses the notion of an \(n\)-relator 3-manifold which is a handlebody with \(n\) 2-handles attached along a system of simple closed curves on its boundary (note that every compact 3-manifold admits such a structure, modulo a completion by 3-balls or 3-handles). It is in the context of \(n\)-relator manifolds that Haken’s theory of normal surfaces in 3-manifolds is generalized. Haken’s theory was based on a fixed chosen handlebody structure of the underlying 3-manifold. In contrast, the present approach is more flexible in allowing changes of the \(n\)-relator structure by slides of the 2-handles (which do not change the homeomorphism type of the underlying 3-manifold). It is shown that every incompressible surface can be isotoped to a surface which is in “strictly normal form” with respect to a set of \(n\)-relator structures, and that the intersection of a strictly normal surface with the boundary of the handlebody from the \(n\)-relator structure can be estimated. This leads naturally to the study of surfaces in handlebodies not intersecting a given system of (attaching) curves (“relative handlebodies”).

The second chapter is devoted to the study of relative handlebodies, after the first one on handlebodies. The book then proceeds, in an organic way of increasing complexity, with the study of the basic and interesting case of 1-relator manifolds, goes on to \(n\)-relator manifolds, and finally comes to “The space of Heegaard graphs” which is the title of the concluding fifth chapter.

Changing the point of view in the above general handle-addition theorem, it can be viewed also as a theorem about Heegaard surfaces (instead of essential surfaces). Then it says that any Heegaard surface can meet a given essential surface in some limited way only. However, “To prove the rigidity theorem concerning Heegaard surfaces it is certainly not enough to consider the intersection of Heegaard surfaces with a sinlge incompressible surface alone; we rather have to study the intersections of Heegaard surfaces, or Heegaard graphs, with a complete hierarchy of surfaces”. This leads, in the final chapter, to the study of the intersections of Heegaard graphs, or more generally, Heegaard 2- complexes, with “Haken 2-complexes” which are associated to hierarchies, culminating in the above rigidity theorem for Heegaard surfaces.

The present is only a very rough outline of the contents of the book. For more details, and in particular for the many (known and new) results and applications proved for various special settings in the course of the development, we refer the reader to the introduction of the book and also to the announcement of the main results in Bull. Am. Math. Soc., New Ser. 23, 91-98 (1990; Zbl 0715.57006)].

The book is quite demanding for the reader. It offers no easy access or shortcuts to its results (the main theorem is on page 420); one has to start more or less from the beginning to get used to its mainly geometric-combinatorial, sometimes quite technical methods, with a strong emphasis on constructive and algorithmic aspects (but as so often “the way is the goal”). However, these remarks apply equally well to Haken’s fundamental work which then turned out to be at the basis of some of the most important theorems in 3-dimensional topology (such as topological rigidity and existence of hyperbolic structures for Haken-3-manifolds).

Reviewer: B.Zimmermann (Trieste)

### 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) |