##
**Lecture notes on generalized Heegaard splittings. Three lectures on low-dimensional topology in Kyoto.**
*(English)*
Zbl 1356.57004

Hackensack, NJ: World Scientific (ISBN 978-981-3109-11-7/hbk; 978-981-3109-13-1/ebook). viii, 130 p. (2016).

This book is for graduate students and researchers in topology. The text addresses generalized Heegaard splittings from the viewpoint of fork complexes.

These notes are well written, definitions are followed by examples, proofs are given in detail. There are exercises included in each chapter. Both, examples and proofs, are depicted with many figures.

In 1898 Heegaard showed that every closed connected compact orientable 3-manifold contains a closed orientable surface which divides the 3-manifold into two handlebodies. This is known as a Heegaard splitting of the 3-manifold.

A. J. Casson and C. McA. Gordon [Topology Appl. 27, 275–283 (1987; Zbl 0632.57010)] generalized this result to connected compact orientable 3-manifolds with non-empty boundary, introducing the concept of compression body, a generalization of handlebody.

M. Scharlemann and A. Thompson [Contemp. Math. 164, 231–238 (1994; Zbl 0818.57013)] extended the notion of Heegaard splitting still further by defining the concept of thin position for a compact 3-manifold. They proceeded as follows: a compact 3-manifold \(M\) can be built in steps; start with a collection of 0-handles or with a submanifold of \(M\), add some 1-handles, then add some 2-handles, then some more 1-handles, then some more 2-handles, etc., so that ultimately: \[ M= M_0 \cup_{F_1} M_1 . . . \cup_{F_n} M_n \] The 1- and 2-handles which occur in \(M_i\) provide it with a Heegaard splitting. This is called a generalized Heegaard splitting of the 3-manifold \(M\). The idea of thin position is to select sets of 1-handles and sets of 2-handles so that the Heegaard splittings are strongly irreducible and the surfaces \(F_i\) are incompressible. To achieve this property, Scharlemann and Thompson defined the width of \(M\), \(w(M)\), as an \((n+1)\)-tuple that encodes the complexity of the Heegaard surfaces for each \(M_i\). A generalized Heegaard splitting that attains the width of \(M\) is called a thin position for \(M\) and satisfies the condition that each Heegaard splitting is strongly irreducible and each \(F_i\) is incompressible. A generalized Heegaard splitting with the above property, but not necessarily realizing the width, is known as a strongly irreducible generalized Heegaard splitting.

Generalized Heegaard splittings and thin position for 3-manifolds have been used as a tool to prove interesting facts in the area of low dimensional topology. Schultens used generalized Heegaard splittings in [J. Schultens, Comment. Math. Helv. 75, No. 3, 353–367 (2000; Zbl 0972.57007)] to give lower bounds for the tunnel number of a connected sum of knots. She used the same tool in [J. Schultens, Geom. Top. 8, 871–876 (2004; Zbl 1055.57023)] to analyze properties of Heegaard splittings of graph manifolds.

Working with thin generalized Heegaard splittings requires to keep account of several properties of the manifold and the decomposition. The authors introduce fork complexes as a tool that accounts for the boundary components of the compression bodies involved in a generalized Heegaard splitting.

The authors develop the topic as follows. In Chapter 1 they give basic definitions such as topological and PL manifolds, and fundamental properties of a surface contained in a 3-manifold.

Chapter 2 introduces Heegaard splittings and several examples such as the Heegaard decomposition for \(S^3\), \(S^2 \times S^1\), lens spaces and knot exteriors.

In Chapter 3 properties of Heegaard splittings are discussed. Using the examples given in Chapter 2 the authors illustrate the properties of being reducible, irreducible, weakly reducible and strongly irreducible, boundary reducible and stabilized. Various lemmas relating these properties are proven. The last part of this section is devoted to prove Haken’s lemma, which includes the Casson and Gordon version for manifolds with non-empty boundary. The technique used for this proof is viewing the Heegaard splittings as a graph in the three dimensional space following the ideas of Scharlemann and Thompson, and Otal.

Waldhausen’s Theorem and the Reidemeister-Singer Theorem are discussed in Chapter 4. To prove these theorems the authors introduce thin position for graphs in the 3-sphere, which is a concept that arises from Morse Theory, this is based on a proof given by Scharlemann and Thompson. The second part of this section gives a demonstration of the Reidemeister-Singer Theorem.

Chapter 5 deals with the definition of fork complex. A 0-fork is obtained by joining a point \(p\) to a point \(g\) with a 1-simplex oriented toward \(g\) and away from \(p\). For \(n\geq1\), an \(n\)-fork is a connected oriented 1-complex which consists of \(n\) distinct points, called tines, which are connected to one point \(p\), called the root, by oriented 1-simplices from the tines to the root, and the root is connected to a point \(g\), called the grip, by a 1-simplex toward \(g\) and away from \(p\).

The tines of an \(n\)-fork represent the non-connected boundary of a compression body, the grip represents the connected boundary of a compression body and the root symbolizes a spine of the compression body.

A fork complex \(\mathcal{F}\) is obtained by attaching two finite collections \(A\) and \(B\) of forks in such a way that a subcollection of tines (grips) of \(A\) is in bijective correspondance with a subcollection of tines (grips) of \(B\). The points of \(\mathcal{F}\) which are not attached to any point form two disjoint subsets denoted by \(\partial_1\mathcal{F}\) and \(\partial_2\mathcal{F}\). It is said that a fork complex \(\mathcal{F}\) is exact if it can be put in \(\mathbb{R}^3\) so that \(\partial_1\mathcal{F}\) lies in the plane of heigth \(0\), \(\partial_2\mathcal{F}\) lies in the plane of heigth 1, and for any path \(\alpha\) in \(\mathcal{F}\) from any point in \(\partial_1\mathcal{F}\) to a point in \(\partial_2\mathcal{F}\), \(h|_\alpha\) is monotonically increasing, where \(h\) is the height function of \(\mathbb{R}^3\).

Given a compact, orientable 3-manifold \(M\) and a partition \((\partial_1 M, \partial_2 M)\) of boundary components of \(M\), the authors define a generalized Heegaard splitting of \((M, \partial_1 M, \partial_2 M)\) as a pair consisting of an exact fork complex \(\mathcal{F}\) and a proper map \(\rho: (M; \partial_1 M, \partial_2 M) \rightarrow (\mathcal{F}; \partial_1\mathcal{F}, \partial_2\mathcal{F})\), so that \(\rho\) satisfies certain conditions. Basically the requirement is that the preimage of a 0-fork \(F \subset \mathcal{F}\) corresponds to a handlebody in \(M\), and the preimage of an \(n\)-fork \(F\subset \mathcal{F}\), with \(n\geq1\), corresponds to a compression body in \(M\), so that if two forks \(F_1, F_2 \subset \mathcal{F}\) are adjacent to a grip \(g\) contained in the interior of \(\mathcal{F}\), then \((\rho^{-1}(F_1), \rho^{-1}(F_2); \rho^{-1}(g))\) is a Heegaard splitting of \(\rho^{-1}(F_1 \cup F_2)\).

Properties of generalized Heegaard splittings are proved via fork complexes. Examples of the fork complex for surface\(\times [0,1]\) and surface\(\times S^1\) are given.

The last two sections of Chapter 5 are devoted to the process of amalgamation which is a way to construct a Heegaard splitting for a 3-manifold \(M\) from a generalized Heegaard splitting, and to the discussion of the boundary stabilization and stabilization problem.

In summary, this book is well written and nicely illustrated, and it is an excellent source for those interested in the theory of Heegaard splittings and generalized Heegaard splittings of compact 3-manifolds.

These notes are well written, definitions are followed by examples, proofs are given in detail. There are exercises included in each chapter. Both, examples and proofs, are depicted with many figures.

In 1898 Heegaard showed that every closed connected compact orientable 3-manifold contains a closed orientable surface which divides the 3-manifold into two handlebodies. This is known as a Heegaard splitting of the 3-manifold.

A. J. Casson and C. McA. Gordon [Topology Appl. 27, 275–283 (1987; Zbl 0632.57010)] generalized this result to connected compact orientable 3-manifolds with non-empty boundary, introducing the concept of compression body, a generalization of handlebody.

M. Scharlemann and A. Thompson [Contemp. Math. 164, 231–238 (1994; Zbl 0818.57013)] extended the notion of Heegaard splitting still further by defining the concept of thin position for a compact 3-manifold. They proceeded as follows: a compact 3-manifold \(M\) can be built in steps; start with a collection of 0-handles or with a submanifold of \(M\), add some 1-handles, then add some 2-handles, then some more 1-handles, then some more 2-handles, etc., so that ultimately: \[ M= M_0 \cup_{F_1} M_1 . . . \cup_{F_n} M_n \] The 1- and 2-handles which occur in \(M_i\) provide it with a Heegaard splitting. This is called a generalized Heegaard splitting of the 3-manifold \(M\). The idea of thin position is to select sets of 1-handles and sets of 2-handles so that the Heegaard splittings are strongly irreducible and the surfaces \(F_i\) are incompressible. To achieve this property, Scharlemann and Thompson defined the width of \(M\), \(w(M)\), as an \((n+1)\)-tuple that encodes the complexity of the Heegaard surfaces for each \(M_i\). A generalized Heegaard splitting that attains the width of \(M\) is called a thin position for \(M\) and satisfies the condition that each Heegaard splitting is strongly irreducible and each \(F_i\) is incompressible. A generalized Heegaard splitting with the above property, but not necessarily realizing the width, is known as a strongly irreducible generalized Heegaard splitting.

Generalized Heegaard splittings and thin position for 3-manifolds have been used as a tool to prove interesting facts in the area of low dimensional topology. Schultens used generalized Heegaard splittings in [J. Schultens, Comment. Math. Helv. 75, No. 3, 353–367 (2000; Zbl 0972.57007)] to give lower bounds for the tunnel number of a connected sum of knots. She used the same tool in [J. Schultens, Geom. Top. 8, 871–876 (2004; Zbl 1055.57023)] to analyze properties of Heegaard splittings of graph manifolds.

Working with thin generalized Heegaard splittings requires to keep account of several properties of the manifold and the decomposition. The authors introduce fork complexes as a tool that accounts for the boundary components of the compression bodies involved in a generalized Heegaard splitting.

The authors develop the topic as follows. In Chapter 1 they give basic definitions such as topological and PL manifolds, and fundamental properties of a surface contained in a 3-manifold.

Chapter 2 introduces Heegaard splittings and several examples such as the Heegaard decomposition for \(S^3\), \(S^2 \times S^1\), lens spaces and knot exteriors.

In Chapter 3 properties of Heegaard splittings are discussed. Using the examples given in Chapter 2 the authors illustrate the properties of being reducible, irreducible, weakly reducible and strongly irreducible, boundary reducible and stabilized. Various lemmas relating these properties are proven. The last part of this section is devoted to prove Haken’s lemma, which includes the Casson and Gordon version for manifolds with non-empty boundary. The technique used for this proof is viewing the Heegaard splittings as a graph in the three dimensional space following the ideas of Scharlemann and Thompson, and Otal.

Waldhausen’s Theorem and the Reidemeister-Singer Theorem are discussed in Chapter 4. To prove these theorems the authors introduce thin position for graphs in the 3-sphere, which is a concept that arises from Morse Theory, this is based on a proof given by Scharlemann and Thompson. The second part of this section gives a demonstration of the Reidemeister-Singer Theorem.

Chapter 5 deals with the definition of fork complex. A 0-fork is obtained by joining a point \(p\) to a point \(g\) with a 1-simplex oriented toward \(g\) and away from \(p\). For \(n\geq1\), an \(n\)-fork is a connected oriented 1-complex which consists of \(n\) distinct points, called tines, which are connected to one point \(p\), called the root, by oriented 1-simplices from the tines to the root, and the root is connected to a point \(g\), called the grip, by a 1-simplex toward \(g\) and away from \(p\).

The tines of an \(n\)-fork represent the non-connected boundary of a compression body, the grip represents the connected boundary of a compression body and the root symbolizes a spine of the compression body.

A fork complex \(\mathcal{F}\) is obtained by attaching two finite collections \(A\) and \(B\) of forks in such a way that a subcollection of tines (grips) of \(A\) is in bijective correspondance with a subcollection of tines (grips) of \(B\). The points of \(\mathcal{F}\) which are not attached to any point form two disjoint subsets denoted by \(\partial_1\mathcal{F}\) and \(\partial_2\mathcal{F}\). It is said that a fork complex \(\mathcal{F}\) is exact if it can be put in \(\mathbb{R}^3\) so that \(\partial_1\mathcal{F}\) lies in the plane of heigth \(0\), \(\partial_2\mathcal{F}\) lies in the plane of heigth 1, and for any path \(\alpha\) in \(\mathcal{F}\) from any point in \(\partial_1\mathcal{F}\) to a point in \(\partial_2\mathcal{F}\), \(h|_\alpha\) is monotonically increasing, where \(h\) is the height function of \(\mathbb{R}^3\).

Given a compact, orientable 3-manifold \(M\) and a partition \((\partial_1 M, \partial_2 M)\) of boundary components of \(M\), the authors define a generalized Heegaard splitting of \((M, \partial_1 M, \partial_2 M)\) as a pair consisting of an exact fork complex \(\mathcal{F}\) and a proper map \(\rho: (M; \partial_1 M, \partial_2 M) \rightarrow (\mathcal{F}; \partial_1\mathcal{F}, \partial_2\mathcal{F})\), so that \(\rho\) satisfies certain conditions. Basically the requirement is that the preimage of a 0-fork \(F \subset \mathcal{F}\) corresponds to a handlebody in \(M\), and the preimage of an \(n\)-fork \(F\subset \mathcal{F}\), with \(n\geq1\), corresponds to a compression body in \(M\), so that if two forks \(F_1, F_2 \subset \mathcal{F}\) are adjacent to a grip \(g\) contained in the interior of \(\mathcal{F}\), then \((\rho^{-1}(F_1), \rho^{-1}(F_2); \rho^{-1}(g))\) is a Heegaard splitting of \(\rho^{-1}(F_1 \cup F_2)\).

Properties of generalized Heegaard splittings are proved via fork complexes. Examples of the fork complex for surface\(\times [0,1]\) and surface\(\times S^1\) are given.

The last two sections of Chapter 5 are devoted to the process of amalgamation which is a way to construct a Heegaard splitting for a 3-manifold \(M\) from a generalized Heegaard splitting, and to the discussion of the boundary stabilization and stabilization problem.

In summary, this book is well written and nicely illustrated, and it is an excellent source for those interested in the theory of Heegaard splittings and generalized Heegaard splittings of compact 3-manifolds.

Reviewer: Fabiola Manjarrez Gutiérrez (Cuernavaca)

### MSC:

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

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