On the cobordism groups of immersions and embeddings

Abstract

In the present paper we suggest as a cobordism invariant of an immersed or embedded submanifold in Euclidean space the singularity set of its projection to a hyperplane. A similar approach has been employed by Banchoff[1] and Koschorke[6]; see also [15]. We consider the range of dimensions n ≤ 3k where n is the dimension and k is the codimension. We prove that in this range (1) our singularity invariant is complete modulo 2-torsion, and (2) modulo-torsion, it can take any value from Thorm's oriented cobordism group of corresponding dimension for k even, while for k odd this invariant is always trivial.