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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