Geometry & Topology

Double point self-intersection surfaces of immersions

Mohammad A Asadi-Golmankhaneh and Peter J Eccles

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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 k1 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+2ΩΣMO(k) of the element απ2k+2ΩΣMO(k) corresponding to the immersion under the Pontrjagin–Thom construction.

Article information

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

Dates
Received: 30 July 1999
Accepted: 29 February 2000
First available in Project Euclid: 21 December 2017

Permanent link to this document
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

Subjects
Primary: 57R42: Immersions
Secondary: 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20] 55Q25: Hopf invariants 57R75: O- and SO-cobordism

Keywords
immersion Hurewicz homomorphism spherical class Hopf invariant Stiefel–Whitney number

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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References

  • M A Asadi-Golmankhaneh, Self-intersection manifolds of immersions, PhD Thesis, University of Manchester (1998)
  • M A Asadi-Golmankhaneh, P J Eccles, Determining the characteristic numbers of self-intersection manifolds, J. London Math. Soc. (to appear)
  • T F Banchoff, Triple points and surgery of immersed surfaces, Proc. Amer. Math. Soc. 46 (1974) 407–413
  • M G Barratt, P J Eccles, $\Gamma^+$–structures III: the stable structure of $\Omega^{\infty}\Sigma^{\infty}A$, Topology, 13 (1974) 199–207
  • J M Boardman, B Steer, On Hopf invariants, Comment. Math. Helv. 42 (1967) 180–221
  • R L W Brown, Immersions and embeddings up to cobordism, Can. J. Math. 23 (1971) 1102–1115
  • R L Cohen, The immersion conjecture for differentiable manifolds, Annals of Math. 122 (1985) 237–328
  • A Dold, Erzeugende der Thomschen Algebra $\mathfrak R$, Math Z. 65 (1956) 25–35
  • E Dyer, R K Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 (1962) 35–88
  • P J Eccles, Multiple points of codimension one immersions of oriented manifolds, Math. Proc. Cambridge Philos. Soc. 87 (1980) 213–220
  • P J Eccles, Codimension one immersions and the Kervaire invariant one problem, Math. Proc. Cambridge Philos. Soc. 90 (1981) 483–493
  • P J Eccles, Characteristic numbers of immersions and self-intersection manifolds, from: “Proceedings of the Colloquium in Topology, Szekszárd, Hungary, August 1993”, Bolyai Society Mathematical Studies 4, James Bolyai Mathematical Society, Budapest (1995) 197–216
  • P J Eccles, Double point manifolds and immersions of spheres in Euclidean space, from “Proceedings of the William Browder Sixtieth Birthday Conference, March 1994”, Annals of Math. Studies 138, Princeton University Press (1997) 125–137
  • S O Kochman, Bordism, stable homotopy and Adams spectral sequences, American Mathematical Society (1996)
  • U Koschorke, B Sanderson, Self intersections and higher Hopf invariants, Topology, 17 (1978) 283–290
  • J Lannes, Sur les immersions de Boy, Lecture Notes in Mathematics 1051, Springer (1984) 263–270
  • J P May, The homology of $E_{\infty}$ spaces, Lecture Notes in Mathematics 533, Springer (1976) 1–68
  • J W Milnor, J D Stasheff, Characteristic classes, Ann. of Math. Studies 76, Princeton University Press (1974)
  • V P Snaith, A stable decomposition of $\Omega^nS^nX$, J. London Math. Soc. 7 (1974) 577–583
  • A Sz\Hucs, Gruppy kobordizmov $l$–pogruženiǐ I, Acta Math. Acad. Sci. Hungar. 27 (1976) 343–358
  • A Sz\Hucs, Gruppy kobordizmov $l$–pogruženiǐ II, Acta Math. Acad. Sci. Hungar. 28 (1976) 93–102
  • A Sz\Hucs, Double point surfaces of smooth immersions $M^n\rightarrow \R^{2n-2}$, Math. Proc. Cambridge Philos. Soc. 113 (1993) 601–613
  • P Vogel, Cobordisme d'immersions, Ann. Sci. Ecole Norm. Sup. 7 (1974) 317–358
  • R Wells, Cobordism groups of immersions, Topology, 5 (1966) 281–294