#### Geometry & Topology, Vol. 4 (2000)
Paper no. 4, pages 149--170.

## Double point self-intersection surfaces of immersions

### Mohammad A Asadi-Golmankhaneh, Peter J Eccles

**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 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.
**Keywords**.
immersion, Hurewicz homomorphism, spherical class, Hopf invariant,
Stiefel-Whitney number

**AMS subject classification**.
Primary: 57R42.
Secondary: 55R40, 55Q25, 57R75.

**DOI:** 10.2140/gt.2000.4.149

**E-print:** `arXiv:math.GT/0003236`

Submitted to GT on 30 July 1999.
Paper accepted 29 February 2000.
Paper published 11 March 2000.

Notes on file formats
Mohammad A Asadi-Golmankhaneh, Peter J Eccles

Department of Mathematics, University of Urmia

PO Box 165, Urmia, Iran

Department of Mathematics, University of Manchester

Manchester, M13 9PL, UK

Email: pjeccles@man.ac.uk

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