2 7 Ju l 2 00 4 A set of moves for Johansson representation

A Dehn sphere Σ [P] in a closed 3-manifold M is a 2-sphere immersed in M with only double curve and triple point singularities. The sphere Σ fills M [Mo2] if it defines a celldecomposition of M . The inverse image in S of the double curves of Σ is the Johansson diagram of Σ [J1] and if Σ 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 [Mo2] 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 [V2] 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.(Math. Subject Classification: 57N10, 57N35)