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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 $f:M\to {ℝ}^{2}$, 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 $q:M\to {W}_{f}$ be the quotient map and $\overline{f}:$W${}_{f}\to {ℝ}^{2}$ be defined by $f=\overline{f}\circ q$. For any stable map $f:M\to {ℝ}^{2}$, they show that the map $\overline{f}:$W${}_{f}\to {ℝ}^{2}$ can be lifted to a map $g:{W}_{f}\to {ℝ}^{4}$ with pleasant properties. By this result, they also give a sufficient condition for existence of a lifting of $f:M\to {ℝ}^{2}$ to an immersion F of M into ${ℝ}^{4}$ (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:
 57R45 Singularities of differentiable mappings 58C25 Differentiable maps on manifolds (global analysis) 58K99 Theory of singularities and catastrophe theory 57R42 Immersions (differential topology) 57N10 Topology of general 3-manifolds