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Oriented bordism groups of immersions

Published online by Cambridge University Press:  24 October 2008

Gui-Song Li
Gui-Song Li
Affiliation:
Institute of Systems Science, Academia Sinica, Beijing 100080, China
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Extract

Let IΩn, k denote the bordism group of immersions of closed oriented n-manifolds into (n + k)-space. The object of this paper is to study certain group extension problems arising from Pastor's calculations of IΩn, k.

The bordism group of immersions was first studied by Wells [12] who calculated the unoriented bordism group I Rn, k for k = n and k = n − 1 ≡ 3(4). Later these unoriented bordism groups were completely determined by Koschorke and Olk for k ≥ n − 2 with the help of an exact sequence measuring the difference between IRn, k and Rn (see [4]). A similar program has been carried out by Pastor [7] to determine the oriented bordism group I Ωn, k for k ≥ n − 2 except for certain group extension problems and some low dimensional cases.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 3 , November 1995 , pp. 467 - 475
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

REFERENCES

[1]Cohen, R. L.. The immersion conjecture for differentiable manifolds. Ann. of Math. 122 (1985), 237–328.CrossRefGoogle Scholar
[2]Hirsch, M. W.. Immersions of manifolds. Trans. Amer. Math. Soc. 93 (1959), 242–276.CrossRefGoogle Scholar
[3]James, I. M.. Some embeddings of projective spaces. Proc. Cambridge Philos. Soc. 55 (1959), 294–298.CrossRefGoogle Scholar
[4]Koschorke, U.. Vector fields and other vector bundle morphisms - a singularity approach. Lecture Notes in Mathematics vol. 847 (Springer-Verlag, 1981).CrossRefGoogle Scholar
[5]Massey, W. S. and Peterson, F. P.. On the dual Stiefel-Whitney classes of a manifold. Bol. Soc. Mat. Mexicana 8 (1963), 1–13.Google Scholar
[6]Olk, C.. Immersionen von Mannigfaltigkeiten in euklidische Räume. Dissertation, Siegen (1980).Google Scholar
[7]Pastor, G.. On bordism groups of immersions. Trans. Amer. Math. Soc. 283 (1984), 295–301.CrossRefGoogle Scholar
[8]Salomonsen, H. A.. On the homotopy groups of Thom complexes and unstable bordism; in Proceedings of Advanced Study Institute in Algebraic Topology (Aärhus, 1970).Google Scholar
[9]Szücs, A.. Cobordism of maps with simplest singularities; in Topology Symposium, Siegen 1979. Lecture Notes in Mathematics vol. 788 (Springer-Verlag, 1980), pp. 223–244.CrossRefGoogle Scholar
[10]Ucci, J. J.. Immersing and embedding Dold manifolds. Topology 4 (1965), 283–293.CrossRefGoogle Scholar
[11]Wall, C. T. C.. Determination of the cobordism ring. Ann. of Math. 72 (1960), 292–311.CrossRefGoogle Scholar
[12]Wells, R.. Cobordism groups of immersions. Topology 5 (1966), 281–294.CrossRefGoogle Scholar