Double point self-intersection surfaces of immersions

Details Double point self-intersection surfaces of immersions Double point self-intersection surfaces of immersions

2000-03-11

arxiv.org/abs/math/0003236v1

Geom. Topol. 4 (2000), 149-170

Mathematics

Geometric Topology

22 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper4.abs.html

Scientific paper

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 is congruent to 1 modulo 4 or k+1 is a power of 2. This corrects a previously published result by Andras Szucs. The method of proof is to evaluate the Stiefel-Whitney numbers of the double point self-intersection surface. By earier work of the authors these numbers can be read off from the Hurewicz image h(\alpha ) in H_{2k+2}\Omega ^{\infty }\Sigma ^{\infty }MO(k) of the element \alpha in \pi _{2k+2}\Omega ^{\infty }\Sigma ^{\infty }MO(k) corresponding to the immersion under the Pontrjagin-Thom construction.

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