STABLE MAPS OF 3-MANIFOLDS INTO THE PLANE AND THEIR QUOTIENT-SPACES

We study stable maps f: M --> R(2) of closed orientable 3-manifolds M into the plane using their quotient spaces, which are defined to be th e spaces of the connected components of f-fibres and which are known t o be 2-dimensional polyhedra. We show that every 3-manifold admits a s table map whose quotient space is homeomorphic to that of a stable map of S-3. We also deduce a Morse-type inequality for stable maps, which implies that there is no universal stable map of S-3 in the above sen se. Certain moves in the quotient spaces corresponding to generic homo topies of stable maps are also studied.