##
**Four-manifolds, geometries and knots.**
*(English)*
Zbl 1087.57015

Geometry and Topology Monographs 5. Coventry: Geometry & Topology Publications. xiv, 379 p. (2002).

This book studies the algebraic and topological characterization of closed 4-manifolds. It collects and extends the author’s work from two previous books [2- knots and their groups, Cambridge University Press (1989; Zbl 0669.57008), The algebraic characterization of geometric 4-manifolds, Cambridge University Press (1994; Zbl 0812.57001)] as well as a large body of his research published in journals and other books. It serves well as a reference for this material and is highly recommended for graduate students and other researchers in the area. Its goal is to characterize algebraically the closed 4-manifolds that fiber nontrivially, admit geometric structures, or are obtained by surgery on 2-knots.

After introducing group theoretic preliminaries in the first chapter, the remainder of the book consists of three parts. The first part (Chapters 2-6) covers “General Results on Homotopy and Surgery”; the second part (Chapters 7-13) covers “Geometries and Geometric Decomposition”; the third part (Chapters 14-18) covers “2-knots”. We briefly survey each part.

In Chapter 2 the author reviews background material on 2-Complexes and \(PD_3\)-complexes. Chapter 3 presents the homotopy theory of \(PD_4\)-complexes, including closed 4-manifolds. These are determined by the algebraic 2-type and orientation character, although in many cases this reduces to the fundamental group, first Stiefel- Whitney class, and Euler characteristic. General criteria are given for two closed 4- manifolds to be homotopy equivalent, and applied to characterize aspherical 4-manifolds. In Chapter 4, mapping tori and circle bundles are studied. The author shows that a finite \(PD_4\)-complex is homotopy equivalent to a mapping torus \(M(f) = X \times I/(x,0) \sim (f(x),1),\) with \(X\) a \(PD_3\)-complex, iff the fundamental group of \(M(f) \) is an extension of the integers by a finitely presentable normal subgroup and the Euler characteristic is 0. He also studies the related problem of characterizing the total spaces of circle bundles over 3-dimensional bases. In particular, he solves this problem in the aspherical case. Chapter 5 studies surface bundles. The author shows that if \(M\) is a \(PD_4\) complex and \(B,F\) are aspherical closed surfaces, then \(M\) is homotopy equivalent to the total space of a bundle with base \(B\) and fiber \(F\) iff the fundamental group of \(M\) is an extension of \(\pi_1(B)\) by \(\pi_1(F)\) and \(\chi(M) = \chi(B)\chi(F).\) In Chapter 6, the Whitehead group and surgery obstruction theory is studied. This is used to show that for certain classes of 4-manifolds, the manifold is determined up to homeomorphism by homotopy type. One such result states that if \(M\) is a closed 4-manifold that is homotopy equivalent to the total space \(E\) of an \(S^2\)-bundle over an aspherical surface, then it is \(s\)-cobordant to \(E\).

The second part studies geometries and geometric decompositions of 4-manifolds. Chapter 7 introduces the 19 4-dimensional geometries and shows the limitations of geometric methods in dimension 4. In Chapter 8 4-dimensional infrasolvmanifolds are characterized up to homeomorphism. The author also studies the question of when such a manifold is the mapping torus by a self-homeomorphism of a 3-manifold. All 4-dimensional infrasolvmanifolds are shown to be determined up to diffeomorphism by their fundamental groups. Chapters 9-12 study the other geometries, organized in terms of the model being homeomorphic to \(\mathbb R^4, S^2\times \mathbb R^2, S^3 \times \mathbb R, \) or being compact. Aspherical geometric 4-manifolds are determined up to \(s\)-cobordism by their homotopy type. The final chapter of this part concerns geometric decompositions of bundle spaces.

The third part concerns 2-knots. Chapter 14 introduces basic constructions related to 2-knots, such as surgery on 2-knots, which often yields aspherical spaces. A main theorem says the 4-manifold resulting from surgery is aspherical iff the knot group is a \(PD_4\)-group. The remaining chapters first treat fundamental group restrictions and then geometric structures that arise for 2-knots, as well as important classes of 2-knots. In Chapter 17 the author characterizes the closed 4-manifolds obtained by surgery on certain 2-knots, and shows that just eight of the 4-dimensional geometries are realized by knot manifolds.

After introducing group theoretic preliminaries in the first chapter, the remainder of the book consists of three parts. The first part (Chapters 2-6) covers “General Results on Homotopy and Surgery”; the second part (Chapters 7-13) covers “Geometries and Geometric Decomposition”; the third part (Chapters 14-18) covers “2-knots”. We briefly survey each part.

In Chapter 2 the author reviews background material on 2-Complexes and \(PD_3\)-complexes. Chapter 3 presents the homotopy theory of \(PD_4\)-complexes, including closed 4-manifolds. These are determined by the algebraic 2-type and orientation character, although in many cases this reduces to the fundamental group, first Stiefel- Whitney class, and Euler characteristic. General criteria are given for two closed 4- manifolds to be homotopy equivalent, and applied to characterize aspherical 4-manifolds. In Chapter 4, mapping tori and circle bundles are studied. The author shows that a finite \(PD_4\)-complex is homotopy equivalent to a mapping torus \(M(f) = X \times I/(x,0) \sim (f(x),1),\) with \(X\) a \(PD_3\)-complex, iff the fundamental group of \(M(f) \) is an extension of the integers by a finitely presentable normal subgroup and the Euler characteristic is 0. He also studies the related problem of characterizing the total spaces of circle bundles over 3-dimensional bases. In particular, he solves this problem in the aspherical case. Chapter 5 studies surface bundles. The author shows that if \(M\) is a \(PD_4\) complex and \(B,F\) are aspherical closed surfaces, then \(M\) is homotopy equivalent to the total space of a bundle with base \(B\) and fiber \(F\) iff the fundamental group of \(M\) is an extension of \(\pi_1(B)\) by \(\pi_1(F)\) and \(\chi(M) = \chi(B)\chi(F).\) In Chapter 6, the Whitehead group and surgery obstruction theory is studied. This is used to show that for certain classes of 4-manifolds, the manifold is determined up to homeomorphism by homotopy type. One such result states that if \(M\) is a closed 4-manifold that is homotopy equivalent to the total space \(E\) of an \(S^2\)-bundle over an aspherical surface, then it is \(s\)-cobordant to \(E\).

The second part studies geometries and geometric decompositions of 4-manifolds. Chapter 7 introduces the 19 4-dimensional geometries and shows the limitations of geometric methods in dimension 4. In Chapter 8 4-dimensional infrasolvmanifolds are characterized up to homeomorphism. The author also studies the question of when such a manifold is the mapping torus by a self-homeomorphism of a 3-manifold. All 4-dimensional infrasolvmanifolds are shown to be determined up to diffeomorphism by their fundamental groups. Chapters 9-12 study the other geometries, organized in terms of the model being homeomorphic to \(\mathbb R^4, S^2\times \mathbb R^2, S^3 \times \mathbb R, \) or being compact. Aspherical geometric 4-manifolds are determined up to \(s\)-cobordism by their homotopy type. The final chapter of this part concerns geometric decompositions of bundle spaces.

The third part concerns 2-knots. Chapter 14 introduces basic constructions related to 2-knots, such as surgery on 2-knots, which often yields aspherical spaces. A main theorem says the 4-manifold resulting from surgery is aspherical iff the knot group is a \(PD_4\)-group. The remaining chapters first treat fundamental group restrictions and then geometric structures that arise for 2-knots, as well as important classes of 2-knots. In Chapter 17 the author characterizes the closed 4-manifolds obtained by surgery on certain 2-knots, and shows that just eight of the 4-dimensional geometries are realized by knot manifolds.

Reviewer: Terry Lawson (New Orleans)

### MSC:

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

57N16 | Geometric structures on manifolds of high or arbitrary dimension |

57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |

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

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