Turkish Journal of Mathematics
DOI

Abstract
The homeomorphism problem is, given two compact nmanifolds, is there an algorithm to decide if the manifolds are homeomorphic or not. The homeomorphism problem has been solved for many important classes of 3manifolds  especially those with embedded 2sided incompressible surfaces (cf [12], [15], [16]), which are called Haken manifolds. It is also wellknown that the homeomorphism problem is easily solvable for two 3manifolds which admit geometries in the sense of Thurston [36], [31]. Hence the recognition problem, to decide if a 3manifold has a geometric structure, is a significant problem. The recognition problem has been solved for all geometric classes, except for the class of small Seifert fibered spaces, which either have finite fundamental group or have fundamental groups which are extensions of Z by a triangle group and have finite abelianisation. Our aim in this paper is to give an algorithm to recognise these last classes of 3manifolds, i.e to decide if a given 3manifold is homeomorphic to one in this class. A completely different solution has been announced recently by Tao Li [22]. Also Perelman's announcement of a solution of the geometrisation conjecture would enable a complete solution of the homeomorphism problem; by identifying which geometric structure a given manifold admits. However it is worth noting that practical algorithms for the homeomorphism and recogntion problems, which can be implemented via software, are very useful for experimentation in 3manifold topology. (See for example [5], [39]).
Keywords
small Seifert fibered space, recognition algorithm, Heegaard splitting, almost normal surface
First Page
75
Last Page
88
Recommended Citation
RUBINSTEIN, J. HYAM (2004) "An Algorithm to Recognise Small Seifert Fiber Spaces," Turkish Journal of Mathematics: Vol. 28: No. 1, Article 5. Available at: https://journals.tubitak.gov.tr/math/vol28/iss1/5