W. Motta et al., STABLE MAPS OF 3-MANIFOLDS INTO THE PLANE AND THEIR QUOTIENT-SPACES, Proceedings of the London Mathematical Society, 71, 1995, pp. 158-174
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.