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On the cobordism groups of immersions and embeddings

Published online by Cambridge University Press:  24 October 2008

András Szücs
András Szücs
Affiliation:
Department of Analysis, Eötvös University, Budapest, Hungary H-1088
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Extract

In the present paper we suggest as a cobordism invariant of an immersed or embedded submanifold in Euclidean space the singularity set of its projection to a hyperplane. A similar approach has been employed by Banchoff[1] and Koschorke[6]; see also [15]. We consider the range of dimensions n ≤ 3k where n is the dimension and k is the codimension. We prove that in this range (1) our singularity invariant is complete modulo 2-torsion, and (2) modulo-torsion, it can take any value from Thorm's oriented cobordism group of corresponding dimension for k even, while for k odd this invariant is always trivial.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 2 , March 1991 , pp. 343 - 349
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Banchoff, T.. Triple points and singularities of projections of smoothly immersed surfaces. Proc. Amer. Math. Soc. 46 (1974), 402–406.CrossRefGoogle Scholar
[2]Barratt, M. G. and Eccles, P.. Γ+-structures I. Topology 13 (1974), 25–45.CrossRefGoogle Scholar
[3]Burlet, O.. Cobordismes de plongements et produits homotopiques. Comment. Math. Helv. 46 (1971), 277–288.CrossRefGoogle Scholar
[4]Golubjatnikov, V.. On a cobordism theory. Sibirskij Mat. Journal 20 (1979), 263–269.Google Scholar
[5]Jänich, K.. Symmetry properties of singularities of C ∞-functions. Math. Ann. 238 (1978), 147–156.CrossRefGoogle Scholar
[6]Koschorke, U.. Vector Fields and Other Vector Bundle Morphisms. Lecture Notes in Math. vol. 847 (Springer-Verlag, 1981).CrossRefGoogle Scholar
[7]Koschorke, U. and Sanderson, B.. Self-intersections and higher Hopf invariants. Topology 17 (1978), 283–290.CrossRefGoogle Scholar
[8]Morin, B.. Formes canoniques des singularités d'une application differentiable. C.R. Acad. Sci. Paris 260 (1965), 5662–5665.Google Scholar
[9]Olk, C.. Immersionen von Mannigfaltigkeiten in euklidische Räume. Dissertation, Universität Siegen (1980).Google Scholar
[10]Pastor, G.. On bordism group of immersions. Proc. Amer. Math. Soc. 283 (1984), 295–301.CrossRefGoogle Scholar
[11]Salomonsen, H. A.. On the homotopy groups of Thorn complexes and unstable bordism. In Proceedings of Advanced Study Institute in Algebraic Topology, 1970 (Aarhus University, 1970).Google Scholar
[12]Stong, R. E.. Notes on Cobordism Theory (Princeton University Press and University of Tokyo Press, 1968).Google Scholar
[13]Szücs, A.. An analogue of the Thom space for mappings with singularity of type Σ1. Mat. Sb. (N.S.) 108 (1979), 433–456.Google Scholar
[14]Szücs, A.. Cobordism of immersions with restricted selfintersections. Osaka J. Math. 21 (1984), 71–80.Google Scholar
[15]Szücs, A.. Cobordism of maps with simplest singularities. In Topology Symposium, Siegen, 1979, Lecture Notes in Math. vol. 788 (Springer-Verlag, 1980), pp. 223–244.CrossRefGoogle Scholar
[16]Szücs, A.. Multiple points of singular maps. Math. Proc. Cambridge Philos. Soc. 100 (1986), 331–346.CrossRefGoogle Scholar
[17]Szücs, A.. Immersions in bordism classes. Math. Proc. Cambridge Philos. Soc. 103 (1988), 89–95.CrossRefGoogle Scholar
[18]Szücs, A.. Universal singular maps. In Proceedings of Topology Conference, Pécs, 1989 (editor Császár, A.), Colloq. Math. Soc. János Bolyai (North-Holland). (Submitted.)Google Scholar
[18]Szücs, A.. To Immersions through Singularities. Regensburger Math. Schriften. (Submitted.)Google Scholar
[19]Wall, C. T. C.. A second note on symmetry of singularities. Bull. London Math. Soc. 12 (1980), 347–357.CrossRefGoogle Scholar
[20]Wells, R.. Cobordism of immersions. Topology 5 (1966), 281–294.CrossRefGoogle Scholar