Double point of self-transverse immersions of M²ⁿ $\looparrowright$ R^4n-5

Abstract

A self-transverse immersion of a smooth manifold M²ⁿ in R^4n-5 for n > 5 has a double point self-intersection set which is the image of an immersion of a smooth 5-dimensional manifold, cobordant to Dold manifold V⁵ or a boundary. We will show that the double point manifold of any such immersion is a boundary. The method of proof is to evaluate the Stiefel-Whitney numbers of the double point self-intersection manifold. By a certain method these numbers can be read off from spherical elements of H_4n-5QMO(2n-5), corresponding to the immersions under the Pontrjagin-Thom construction.