Stable maps of 3-manifolds into the plane and their quotient spaces

We study stable maps f: M → R²of closed orientable 3-manifolds M into the plane using their quotient spaces, which are defined to be the spaces of the connected components of f-fibres and which are known to be 2-dimensional polyhedra. We show that every 3-manifold admits a stable map whose quotient space is homeomorphic to that of a stable map of S³We also deduce a Morse-type inequality for stable maps, which implies that there is no universal stable map of S³in the above sense. Certain moves in the quotient spaces corresponding to generic homotopies of stable maps are also studied.