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 2 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) *
A filling Dehn sphere Σ in a closed 3-manifold M is a sphere transversely immersed in M that defines a cell decomposition of M . Every closed 3-manifold has a filling Dehn sphere [9]. The Montesinos complexity of a 3-manifold M is defined as the minimal number of triple points among all the filling Dehn spheres of M . A sharp upper bound for the Montesinos complexity of the connected sum of two 3-manifolds is given.
The triple point numbers and the triple point spectrum of a closed 3-manifold were defined in [14]. They are topological invariants that give a measure of the complexity of a 3-manifold using the number of triple points of minimal filling Dehn surfaces. Basic properties of these invariants are presented, and the triple point spectra of S 2 × S 1 and S 3 are computed.2000 Mathematics Subject Classification. Primary 57N10, 57N35.
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