On the multiple points of immersions in Euclidean spaces

Abstract

Abstract. Given a self-transverse immersion of a closed, oriented manifold in a euclidean space and a natural number i we compute the oriented cobordism class of the manifold of i-tuple points. §0. IntroductionLet f : M n → R n+k be a generic smooth immersion of a closed smooth manifold M n in the euclidean space R n+k and let ∆ i (f ) denote the manifold of i-fold points in the image. The manifold ∆ i (f ) is immersed in R n+k by a (non-generic) immersion g i ; the image of the non-multiple points of g i is the set of those points in R n+k which have exactly i preimages.The aim of this paper is to investigate the topological characteristics of these manifolds ∆ i (f ). In particular we compute their signatures and more generally all their Pontrjagin numbers, show that in many cases they have even Euler characteristics, and investigate whether the cobordism class of the manifold M n determines their cobordism class or not. It turns out that when the multiplicity i is odd, then the oriented cobordism class [∆ i (f )] of ∆ i (f ) is determined by the oriented cobordism class of M n , while for i even the oriented cobordism class [∆ i (f )] is independent of the cobordism class of M n . §1. Formulation of the resultsThe prototype of this kind of result is that of Banchoff: