A set of moves for Johansson representation of 3-manifolds. An outline

Abstract

A Dehn sphere in a closed 3-manifold M is a 2-sphere immersed in M with only double curve and triple point singularities. The Dehn sphere S fills M if it defines a cell-decomposition of M. The inverse image in S^{2} of the double curves of S is the Johansson diagram of S and if S fills M it is possible to reconstruct M from the diagram. A Johansson representation of M is the Johansson diagram of a filling Dehn sphere of M. In a recent paper of J. M. Montesinos it is proved that every closed 3-manifold has a Johansson representation coming from a nulhomotopic filling Dehn sphere. In this paper a set of moves for Johansson representations of 3-manifolds is given. In a forthcoming paper it is proved that this set of moves suffices for relating different Johansson representations of the same 3-manifold coming from nulhomotopic filling Dehn spheres. The proof of this result is outlined here.