Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3-dimensional manifolds. (English) Zbl 0889.57021

Rubinstein, J. H.

Kazez, William H. (ed.), Geometric topology. 1993 Georgia international topology conference, August 2–13, 1993, Athens, GA, USA. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 2(pt.1), 1-20 (1997).

The homeomorphism problem for 3-manifolds can be formulated as follows: Is there an algorithm to decide whether or not two given 3-manifolds are homeomorphic? (A special case of this is the so-called recognition problem: is there an algorithm to decide whether a given 3-manifold is homeomorphic to the 3-sphere?)
In this paper, the author gives a partial answer to the homeomorphism problem; he proves the following theorem:
Theorem 2: There is an algorithm to find the minimal Heegaard genus of a triangulated closed orientable irreducible 3-manifold. Moreover there is an algorithm to construct all minimal genus Heegaard splittings of such a manifold (up to isotopy and Dehn twisting about embedded incompressible tori).
As a corollary, this implies that the homeomorphism problem does have a positive answer in certain cases:
Corollary 1: There is an algorithm to solve the homeomorphism problem for 3-manifolds which are closed, triangulated, orientable, irreducible and have Heegaard genus at most two.
The theorem is proved by techniques inspired by W. Haken [Stud. Math. 5 (Studies Modern Topol.), 39-98 (1968; Zbl 0194.24902)], the main tool are “normal” and, slightly more general, “almost normal surfaces”. Along the way of the proof, the author proves two further results which are of interest in themselves:
As theorem 1, he outlines a proof of the recognition problem for the 3-sphere. As theorem 3, he shows that under the conditions of theorem 2, any strongly irreducible Heegaard splitting can be isotoped to an almost normal surface.
The rest of the proof consists, roughly speaking, in showing that a family of surfaces can be constructed containing any surface coming from a Heegaard splitting up to a certain genus.
For the entire collection see [Zbl 0882.00042].

Reviewer: Alexander Frey (Berlin)

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57N12	Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57M40	Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)

Keywords:

Heegaard genus; Heegaard splitting; normal surface; Haken techniques; recognition problem; homeomorphism problem; decision problem

Citations:

Zbl 0194.24902