Existence problem for fold maps

This is a survey article on the existence problem of fold maps. Let M-n be a closed n-manifold and f : M-n -> R-p a generic smooth map into the p-dimensional Euclidean space with n >= p. If f has only fold singularities as its singularities, then f is called a fold map. When p = 1, a fold map is nothing but a Morse function. Then we will consider a problem belonging to global singularity theory: Find the necessary and/or sufficient condition(s) for the existence of fold maps. In the problem for the cases where p = 3 and p = 7, the existence of fold maps has a special position if we assume that the Euler characterisitc of the source manifold should be odd. The main purpose of the paper is to discuss the problem in the case that p = 3, 7 respectively by summarizing recent results in relation with the Thom polynomials, the Eliashberg-Ando h-principle theorem, etc. We will also give a new result on the necessary and sufficient condition for the existence problem of fold maps of odd dimensional manifolds into R-3.