Double point of self-transverse immersions of ${M^{2n} \looparrowright \boldsymbol{R}^{4n-5}}$

Abstract

A self-transverse immersion of a smooth manifold $M^{2n}$ in $\boldsymbol{R}^{4n-5}$ for $n \gt 5$ has a double point self-intersection set which is the image of an immersion of a smooth 5-dimensional manifold, cobordant to Dold manifold $V^5$ 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 $H_{4n-5}QMO(2n-5)$, corresponding to the immersions under the Pontrjagin-Thom construction.