## 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.

### 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

### Citations:

Zbl 0669.57008; Zbl 0812.57001
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