Let f : M → Rp be a smooth map of a closed n-dimensional manifold M into Rp (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 R3 and a classification theorem of 4- and 5-dimensional manifolds with free fundamental groups admitting special generic maps into R3. Furthermore, we study the topological structure of the surfaces which arise as the singular set of a special generic map into R3 on a given manifold.
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