A Dehn sphere Σ in a closed 3-manifold M is a 2-sphere immersed in M with only double curve and triple point singularities. The Dehn sphere Σ fills M if it defines a cell decomposition of M. The inverse image in S² of the double curves of Σ is the Johansson diagram of Σ 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. 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm190-0-10,
author = {Rub\'en Vigara},
title = {A set of moves for Johansson representation of 3-manifolds},
journal = {Fundamenta Mathematicae},
volume = {189},
year = {2006},
pages = {245-288},
zbl = {1100.57023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm190-0-10}
}
Rubén Vigara. A set of moves for Johansson representation of 3-manifolds. Fundamenta Mathematicae, Tome 189 (2006) pp. 245-288. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm190-0-10/