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Abstract

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