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Double point self-intersection surfaces of immersions

Mohammad A Asadi-Golmankhaneh and Peter J Eccles

Geometry & Topology 4 (2000) 149–170

arXiv: math.GT/0003236

Abstract

A self-transverse immersion of a smooth manifold Mk+2 in 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 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) H2k+2ΩΣMO(k) of the element α π2k+2ΩΣMO(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
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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