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Mapping three-manifolds into the plane. I. (English) Zbl 0586.57018
In this paper the authors study global properties of stable maps of compact three manifolds into the plane. Let M be a compact orientable 3- manifold without boundary. For a stable map f:M 2 , let W f be the quotient of M obtained by identifying two points of M if they are in the same connected component of the same fibre of f. Let q:MW f be the quotient map and f ¯:W f 2 be defined by f=f ¯q. For any stable map f:M 2 , they show that the map f ¯:W f 2 can be lifted to a map g:W f 4 with pleasant properties. By this result, they also give a sufficient condition for existence of a lifting of f:M 2 to an immersion F of M into 4 (Theorem 2.2). This result is a very interesting result in the same direction as A. Haefliger [Ann. Inst. Fourier 10, 47-60 (1960; Zbl 0095.377)] and Y. Saito [J. Math. Kyoto Univ. 1, 425-455 (1962; Zbl 0201.563)].
Reviewer: S.Izumiya

MSC:
57R45Singularities of differentiable mappings
58C25Differentiable maps on manifolds (global analysis)
58K99Theory of singularities and catastrophe theory
57R42Immersions (differential topology)
57N10Topology of general 3-manifolds