Topology of definite fold singularities

It is known as the Reeb theorem that if a closed  differentiable manifold admits a smooth
function with only minima and  maxima as its critical points, then the manifold is necessarily 
homeomorphic to the sphere. In this talk some generalizations of this  theorem will be
presented for smooth maps into higher dimensional  Euclidean spaces. Unlike the function
case, the existence of such maps  strongly affects the differentiable structure of the manifold, 
especially in dimension four.