An algorithm to recognize the 3-sphere. (English) Zbl 0864.57009

Chatterji, S. D. (ed.), Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 601-611 (1995).
The author announces a number of important and far-reaching results in the theory of 3-manifolds. Much of it based on polyhedral minimal surface techniques developed in joint work of the author with J. T. Pitts. The results include the solution of the famous and longstanding recognition problem for the 3-sphere. Published versions of Rubinstein’s algorithm are given by A. Thompson [Math. Res. Lett. 1, 613-630 (1994; Zbl 0849.57009)] and S. V. Matveev [Sb. Math. 186, No. 5, 695-710 (1995); translation from Mat. Sb. 186, No. 5, 69-84 (1995; Zbl 0849.57010)]. It is also announced that the set of all Heegaard surfaces of given genus is finite for all closed, orientable 3-manifolds up to ambient homeomorphisms (the case of Haken 3-manifolds has been considered earlier by the reviewer in [Topology and combinatorics of 3-manifolds, Lect. Notes Math. 1599 (1995; Zbl 0820.57001)]. It is further announced that the techniques leading to the previous results can be combined with Hatcher’s solution of the Smale conjecture in order to prove that any finite group that acts freely on the 3-sphere admits a free, orthogonal action on the 3-sphere, thereby extending greatly a well-known theorem by F. Waldhausen [Topology 8, 81-91 (1969; Zbl 0185.27603)] and an earlier announcement by the author.
For the entire collection see [Zbl 0829.00014].


57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)