#### Volume 4, issue 1 (2000)

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Double point self-intersection surfaces of immersions

Geometry & Topology 4 (2000) 149–170
 arXiv: math.GT/0003236
##### 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 $k\equiv 1\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}4$ 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.

##### Keywords
immersion, Hurewicz homomorphism, spherical class, Hopf invariant, Stiefel–Whitney number
##### Mathematical Subject Classification 2000
Primary: 57R42
Secondary: 55R40, 55Q25, 57R75