Double point self-intersection surfaces of immersions

Abstract

A self-transverse immersion of a smooth manifold $M^{k+2}$ in ${ℝ}^{2k+2}$ has a double point self-intersection set which is the image of an immersion of a smooth surface, the double point self-intersection surface. We prove that this surface may have odd Euler characteristic if and only if $k\equiv 1mod4$ or $k+1$ is a power of 2. This corrects a previously published result by András Szűcs.

The method of proof is to evaluate the Stiefel–Whitney numbers of the double point self-intersection surface. By an earlier work of the authors, these numbers can be read off from the Hurewicz image $h\left(\alpha \right)\in H_{2k+2}{\Omega }^{\infty }{\Sigma }^{\infty }MO\left(k\right)$ of the element $\alpha \in {\pi }_{2k+2}{\Omega }^{\infty }{\Sigma }^{\infty }MO\left(k\right)$ corresponding to the immersion under the Pontrjagin–Thom construction.