An algorithm to find vertical tori in small Seifert fiber spaces (Q851098)

One of the general problems of \(3\)-manifold theory is to give algorithms to recognize topological properties from a given combinatorial structure, such as a triangulation or a Heegaard decomposition. In particular, an algorithm to decide whether an irreducible \(3\)-manifold is Haken was given by \textit{W. Jaco} and \textit{U. Oertel} [Topology 23, 195--209 (1984; Zbl 0545.57003)], and one to decide whether it is a lens space was given by \textit{J. H. Rubinstein} [Geometric topology. 1993 Georgia international topology conference, August 2--13, 1993, Athens, GA, USA, Stud. Adv. Math. 2(pt.1), 1--20 (1997; Zbl 0889.57021)], the latter also leading to an algorithm to recognize a small Seifert fiber space with finite fundamental group. The main result of the paper under review is an algorithm to recognize Seifert fiber spaces. Other algorithms to do this have also been given by \textit{J. H. Rubinstein} [Turk. J. Math. 28, No. 1, 75--87 (2004; Zbl 1061.57023)] and \textit{I. Agol}. None of the latter three algorithms is efficient, although the author remarks that his algorithm is easier to implement and he suspects is faster as well. An outline of the author's approach is as follows. Since there is an algorithm to recognize Haken manifolds and Seifert-fibered Haken manifolds [\textit{W. Jaco} and \textit{J. L. Tollefson}, Ill. J. Math. 39, No. 3, 358--406 (1995; Zbl 0858.57018)], one need consider only the small case, and applying Rubinstein's algorithms, it is only necessary to recognize small Seifert fibered spaces with infinite fundamental group. P. Scott proved, among other things, that such a manifold always contains a vertical immersed essential torus which has only one or two double curves, and has the \(4\)-plane property (the preimage planes in the universal cover have the property that there is a disjoint pair among any four of them). Using Scott's setup, the author shows that such a torus has the \(7\)-color property, which is that the preimage planes can be colored so that no two of the same color intersect. The next observation is of independent interest: given an \(n\) and a compact \(3\)-manifold \(M\), there is a finite collection of immersed branched surfaces such that any immersed \(\pi_1\)-injective surface in \(M\) with the \(n\)-color property is carried by some member of the collection. The author also develops some additional properties of immersed branched surfaces. The next step is to show how to determine whether \(M\) is a small Seifert fibered space of Euclidean type (i.e. the quotient orbifold \(X\) has orbifold Euler characteristic equal to \(0\)). The last step is to show that if \(M\) is a small Seifert fiber space of hyperbolic type (i.e. \(X\) has negative orbifold Euler characteristic), then there is an algorithmically calculable number \(W\) such that one of the branched surfaces carries an immersed essential torus with weight at most~\(W\), so that searching for them is a finite algorithmic process.