Topology of special generic maps of manifolds into Euclidean spaces

In this paper we study the global topology of special generic maps; i.e., smooth maps of closed n-manifolds into R^p (p<n) all of whose singularities are the definite fold points. Associated with a special generic map is the Stein factorization introduced in a paper of Burlet and de Rham, which is the space of the connected components of the fibers of the map. Our key idea is to reconstruct every special generic map from the Stein factorization, which enables us to obtain various topological restrictions imposed on the source manifolds and the singular sets. When p = 2, and when p = 3 and the source manifolds are 1-connected, we determine the diffeomorphism types of those manifolds which admit special generic maps into R^p, extending results of Burlet and de Rham and Porto and Furuya.