Let f : M → R p be a smooth map of a closed n-dimensional manifold M into R p (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 3 and a classification theorem of 4-and 5-dimensional manifolds with free fundamental groups admitting special generic maps into R 3 . Furthermore, we study the topological structure of the surfaces which arise as the singular set of a special generic map into R 3 on a given manifold.

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Abstract. Let f : M → N be a generic smooth map with corank one singularities between manifolds, and let S(f ) be the singular point set of f . We define the self-intersection class I(S(f )) ∈ H * (M ; Z) of S(f ) using an incident class introduced by Rimányi but with twisted coefficients, and give a formula for I(S(f )) in terms of characteristic classes of the manifolds. We then apply the formula to the existence problem of fold maps.