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Immersions in bordism classes

Published online by Cambridge University Press:  24 October 2008

András Szücs
András Szücs
Affiliation:
Department of Analysis I, Eötvös University, Budapest, Hungary
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Extract

1·1. The aim of the present paper is to prove the following

Theorem. Let Vi and Mn be closed smooth manifolds and i < n. Let f: Vi → Mn be a generic smooth map such that all of its singular points are of the type Σ;1, 0. (See [2].) Then there exists a non-zero integer N such that the bordism class N·[f]εΩi(Mn) contains an immersion.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 1 , January 1988 , pp. 89 - 95
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Ando, Y.. On the elimination of Morin's singularities. J. Math. Soc. Japan 37 (1985), 471–488.CrossRefGoogle Scholar
[2]Boardman, J. M.. Singularities of differentiable maps. I.H.E.S. Publ. Math. 33 (1967), 21–57.CrossRefGoogle Scholar
[3]Brown, E. H.. Cohomology theories. Ann. Math. 75 (1962), 467–484CrossRefGoogle Scholar
Brown, E. H.. Cohomology theories. Ann. Math. 90 (1969), 157–186.Google Scholar
[4]Buoncristiano, S. and Dedo, M.. On resolving singularities and relating bordism to homology. Ann. Sc. Norm. Sup. Pisa (4) 7 (1980), 605–624.Google Scholar
[5]Burlet, O.. Cobordismes de plongements et produits homotopiques. Comment. Math. Helv. 46 (1971), 277–289.CrossRefGoogle Scholar
[6]Conner, P. E. and Floyd, E. E.. Differentiable Periodic Maps. Ergebnisse der Math. Band 33 (Springer-Verlag, 1964).Google Scholar
[7]du Plessis, A. A.. Maps without certain singularities. Comment. Math. Helv. 50 (1975), 363–382.CrossRefGoogle Scholar
[8]Eliashberg, J. M.. On singularities of folding type. Math. USSR Isvestija 4 (1970), 1119–1134.CrossRefGoogle Scholar
[9]Ellis, D.. Unstable bordism groups and isolated singularities. Trans. Amer. Math. Soc. 274 (1982), 695–708.CrossRefGoogle Scholar
[10]Koschorke, U.. Vector fields and other vector bundle morphisms. Lecture Notes in Math. vol. 847 (Springer-Verlag, 1981).CrossRefGoogle Scholar
[11]Morin, B.. Formes canoniques des singularités d'une application différentiable. C. R. Acad. Sci. Paris 260 (1965), 5662–5665.Google Scholar
[12]Pastor, G.. On bordism groups of immersions. Trans. Amer. Math. Soc. 283 (1984), 295–301.CrossRefGoogle Scholar
[13]Serre, J. P.. Homologie singulière des espaces fibrés. Appl. Ann. Math. 54 (1951), 425–505.Google Scholar
[14]Szücs, A.. Cobordisms of maps with simplest singularities. In Topology Symposium, Siegen 1979. Lecture Notes in Math. vol. 788 (Springer-Verlag 1980), 223–244.CrossRefGoogle Scholar
[15]Szücs, A.. Cobordism of immersions with restricted selfintersections. Osaka J. Math. 21 (1984), 71–80.Google Scholar
[16]Szücs, A.. Multiple points of singular maps. Math. Proc. Cambridge Philos. Soc. 100 (1986), 331–346.CrossRefGoogle Scholar
[17]Szücs, A.. Applications of the Pontrjagin–Thom construction for singular maps. Periodica Mathematica Hungarica. (Submitted.)Google Scholar
[18]Thom, R.. Quelques propriétés globales des variétés differentiables. Comment. Math. Helv. 28 (1954), 17–86.CrossRefGoogle Scholar
[19]Wells, R.. Cobordism groups of immersions. Topology 5 (1966), 281–294.CrossRefGoogle Scholar