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

$f:M\to {\mathbb{R}}^{2}$, 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

${}_{f}\to {\mathbb{R}}^{2}$ be defined by

$f=\overline{f}\circ q$. For any stable map

$f:M\to {\mathbb{R}}^{2}$, they show that the map

$\overline{f}:$W

${}_{f}\to {\mathbb{R}}^{2}$ can be lifted to a map

$g:{W}_{f}\to {\mathbb{R}}^{4}$ with pleasant properties. By this result, they also give a sufficient condition for existence of a lifting of

$f:M\to {\mathbb{R}}^{2}$ to an immersion F of M into

${\mathbb{R}}^{4}$ (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)].