Algorithmic classification of 3-manifolds and knots (Q2735923)

The paper is concerned with an important problem in the topology of compact 3-manifolds, known in the literature as the recognition problem for 3-manifolds, that is: does there exist an algorithm to decide whether or not two given 3-manifolds are homeomorphic? The importance of this problem arises from the fact that the positive answer would imply the existence of an algorithmic classification of 3-manifolds. This problem has a positive solution for the class of sufficiently large 3-manifolds, i.e., the manifolds which contain a two-sided closed incompressible surface (different from the 2-sphere and the real projective plane). Examples of such 3-manifolds are given by knot complement spaces (in the 3-sphere), so the above-mentioned positive solution implies the existence of an algorithmic classification for knots. The author observes that all papers and books devoted to the recognition problem for sufficiently large 3-manifolds [see for example \textit{G. Hemion}, Acta Math. 142, 123-155 (1979; Zbl 0402.57027); The classification of knots and 3-dimensional spaces (1992; Zbl 0771.57001); \textit{K. Johannson}, Jahresber. Dtsch. Math.-Ver. 86, No. 2, 37-68 (1984; Zbl 0542.57002); Topology and combinatorics of 3-manifolds, Lect. Notes Math. 1599 (1995; Zbl 0820.57001); \textit{F. Waldhausen}, Proc. Symp. Pure Math., Vol. 32, Part 2, 21-38 (1978; Zbl 0391.57011)] do not contain a complete proof. In this paper the author fills the gap by describing a modified proof based on the same ideas (in particular, the Haken theory of normal surfaces), but the new proof is much shorter and simpler than the original one. The paper is based on a talk given by the author at Caen University during a period of visiting professor at IHES in January 1999. I recommend this paper to any researcher in the topology of compact 3-manifolds.