On the singularities of hyperplane projections of immersions

Given an oriented manifold and its immersion in a euclidean space, we compute the oriented cobordism class of the manifold of Sigma(1r) singular points of the projection of the immersion to a hyperplane. For immersions of non-oriented manifolds, we show that the cobordism class of the domain manifold determines those of all Sigma(1r) singularity manifolds of the hyperplane projection. Finally, we investigate the possible (algebraic) number of cusps (that is, Sigma(1,1) singular points) of generic maps of oriented 4t-manifolds in R(6t-1).