A modification of the Rubinstein-Thompson criterion for a 3-manifold to be the 3-sphere is proposed. Special cell decompositions, called Q-triangulations and irreducible Q-triangulations, for closed compact orientable 3-manifolds are introduced. It is shown that if a closed compact orientable 3-manifold M3 is given by a triangulation (or by a Q-triangulation) then one can effectively decompose M3 into a connected sum of finitely many 3-manifolds some of which are given by irreducible Q-triangulations and others are 2-sphere bundles over a circle. Furthermore, it is shown that the problem whether a 3-manifold given by an irreducible Q-triangulation is homeomorphic to the 3-sphere is in NP, and the problem whether a Q-triangulation of a 3-manifold is irreducible is in coNP.
ASJC Scopus subject areas