## Geometry & Topology

### Double point self-intersection surfaces of immersions

#### Abstract

A self-transverse immersion of a smooth manifold $Mk+2$ in $ℝ2k+2$ 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 $k≡1 mod4$ 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 $h(α)∈H2k+2Ω∞Σ∞MO(k)$ of the element $α∈π2k+2Ω∞Σ∞MO(k)$ corresponding to the immersion under the Pontrjagin–Thom construction.

#### Article information

Source
Geom. Topol., Volume 4, Number 1 (2000), 149-170.

Dates
Accepted: 29 February 2000
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883281

Digital Object Identifier
doi:10.2140/gt.2000.4.149

Mathematical Reviews number (MathSciNet)
MR1742556

Zentralblatt MATH identifier
0941.57025

#### Citation

Asadi-Golmankhaneh, Mohammad A; Eccles, Peter J. Double point self-intersection surfaces of immersions. Geom. Topol. 4 (2000), no. 1, 149--170. doi:10.2140/gt.2000.4.149. https://projecteuclid.org/euclid.gt/1513883281

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