On special generic maps into R3

Abstract

Let f : M → R be a smooth map of a closed n-dimensional manifold M into R (n ≥ p) which has only definite fold singularities as its singular points. Such a map is called a special generic map, which was first defined by Burlet and de Rham for (n, p) = (3, 2) and later extended to general (n, p) by Porto, Furuya, Sakuma and Saeki. In this paper, we study the global topology of such maps for p = 3 and give various new results, among which are a splitting theorem for manifolds admitting special generic maps into R and a classification theorem of 4and 5-dimensional manifolds with free fundamental groups admitting special generic maps into R. Furthermore, we study the topological structure of the surfaces which arise as the singular set of a special generic map into R on a given manifold.