Double point self-intersection surfaces of immersions

A self-transverse immersion of a smooth manifold $M^{k + 2}$ in $ℝ^{2 k + 2}$ 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 $k \equiv 1 mod 4$ 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 $h (α) \in H_{2 k + 2} Ω^{\infty} Σ^{\infty} M O (k)$ of the element $α \in π_{2 k + 2} Ω^{\infty} Σ^{\infty} M O (k)$ 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

References

Forward citations

Publication

Received: 30 July 1999

Accepted: 29 February 2000

Published: 11 March 2000

Proposed: Ralph Cohen

Seconded: Gunnar Carlsson, Yasha Eliashberg

Authors

Mohammad A Asadi-Golmankhaneh
Department of Mathematics University of Urmia PO Box 165 Urmia Iran

Peter J Eccles
Department of Mathematics University of Manchester Manchester M13 9PL United Kingdom

Volume 4, issue 1 (2000)

Mohammad A Asadi-Golmankhaneh and Peter J Eccles