##
**The classification of knots and 3-dimensional spaces.**
*(English)*
Zbl 0771.57001

Oxford Science Publications. Oxford: Oxford University Press. 163 p. (1992).

The classical knot problem was to find an algorithm to determine whether two knots in the 3-sphere are equivalent. More generally, given two Haken 3-manifolds (with boundary patterns), is there an algorithm to determine whether they are homeomorphic (respecting the boundary patterns)? In 1979 the author completed the last step of a program of W. Haken, thereby proving the existence of the algorithm. The goal of the book under review is to give an exposition of this algorithm which is accessible to readers who have only a basic knowledge of point set topology.

After some introductory motivation and background material (such as the statement of the classification of 2-manifolds and a discussion of PL transversality), the author develops some theory of normal surfaces and sets up the main induction for the algorithm. The key idea is that there is a finite family of normal surfaces which can be used to represent all normal surfaces (because any system of linear Diophantine equations has finite families of fundamental solutions which span the set of all solutions). This implies that the 3-manifold can contain only finitely many incompressible surfaces of less than a given complexity (measured by a formula involving Euler characteristic and intersection with the boundary pattern) and an algorithm that finds the surfaces in the second manifold of less than the complexity of one that exists in the first 3- manifold must terminate. One then splits the first manifold along its surface and the second along each of the finitely many surfaces in the second, and tests them inductively; the induction terminates because of the fact that suitable sequences of splittings (hierarchies) must terminate with components that are 3-balls with boundary patterns, which can be mechanically compared. Surfaces of small Euler characteristic create some complications in carrying out this method.

After describing this procedure, the author points out that it is not quite enough, since even though the split manifolds may be homeomorphic, there might be infinitely many ways that the surfaces may be reglued to recover the original manifolds. By reselecting the definition of complexity, one may ensure in most cases that there are only finitely many ways of reconstructing the original manifold. But in the “difficult case” of 3-manifolds which fiber over the circle, the classification amounts to finding an algorithm for classifying (up to conjugacy) the attaching homeomorphisms for the 2-dimensional fiber. This is the last step of Haken’s program, which was completed by the author’s article [On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds, Acta Math. 142, 123-155 (1979; Zbl 0402.57027)], reprinted in the book under review.

After some introductory motivation and background material (such as the statement of the classification of 2-manifolds and a discussion of PL transversality), the author develops some theory of normal surfaces and sets up the main induction for the algorithm. The key idea is that there is a finite family of normal surfaces which can be used to represent all normal surfaces (because any system of linear Diophantine equations has finite families of fundamental solutions which span the set of all solutions). This implies that the 3-manifold can contain only finitely many incompressible surfaces of less than a given complexity (measured by a formula involving Euler characteristic and intersection with the boundary pattern) and an algorithm that finds the surfaces in the second manifold of less than the complexity of one that exists in the first 3- manifold must terminate. One then splits the first manifold along its surface and the second along each of the finitely many surfaces in the second, and tests them inductively; the induction terminates because of the fact that suitable sequences of splittings (hierarchies) must terminate with components that are 3-balls with boundary patterns, which can be mechanically compared. Surfaces of small Euler characteristic create some complications in carrying out this method.

After describing this procedure, the author points out that it is not quite enough, since even though the split manifolds may be homeomorphic, there might be infinitely many ways that the surfaces may be reglued to recover the original manifolds. By reselecting the definition of complexity, one may ensure in most cases that there are only finitely many ways of reconstructing the original manifold. But in the “difficult case” of 3-manifolds which fiber over the circle, the classification amounts to finding an algorithm for classifying (up to conjugacy) the attaching homeomorphisms for the 2-dimensional fiber. This is the last step of Haken’s program, which was completed by the author’s article [On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds, Acta Math. 142, 123-155 (1979; Zbl 0402.57027)], reprinted in the book under review.

Reviewer: D.McCullough (Norman)

### MSC:

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

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

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

57M99 | General low-dimensional topology |