Recognizing the 3-sphere

Abstract

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 $M^3$ is given by a triangulation (or by a $Q$-triangulation) then one can effectively decompose $M^3$ 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 $\textup{\textbf{NP}}$, and the problem whether a $Q$-triangulation of a 3-manifold is irreducible is in $\textup{\textbf{coNP}}$.