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Immersions with non-zero normal vector fields

Published online by Cambridge University Press:  24 October 2008

Bang-He Li  and
Gui-Song Li
Bang-He Li
Affiliation:
Institute of Systems Science, Academia Sinica, Beijing 100080, China
Gui-Song Li
Affiliation:
Institute of Systems Science, Academia Sinica, Beijing 100080, China
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Let M be a smooth n-manifold, X be a smooth (2n − 1)-manifold, and g:M → X be a map. It was proved in [6] that g is always homotopic to an immersion. The set of homotopy classes of monomorphisms from TM into g*TX, which is denoted by Sg, may be enumerated either by the method of I. M. James and E. Thomas or by the singularity method of U. Koschorke (see [1] and references therein). When the natural action of π1(XM, g) on Sg is trivial, for example, if X is euclidean, the set Sg is in one-to-one correspondence with the set of regular homotopy classes of immersions homotopic to g (see e.g. [4]).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 2 , September 1992 , pp. 281 - 285
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

[1]Kaiser, U. and Li, B.-H.. Enumeration of immersions of m-manifolds in (2m−2)-manifolds by the singularity method. In Proceedings of the 1987 Siegen Topology Conference, Lecture Notes in Math. vol. 1350 (Springer-Verlag, 1988), pp. 171–187.Google Scholar
[2]Koschorke, U.. Vector Fields and Other Vector Bundle Morphisrns – a Singularity Approach. Lecture Notes in Math. vol. 847 (Springer-Verlag, 1981).Google Scholar
[3]Koschorke, U.. The singularity method and immersions of m-manifolds into manifolds of dimension 2m−2, 2m−3, 2m−4. In Proceedings of the 1987 Siegen Topology Conference, Lecture Notes in Math. vol. 1350 (Springer-Verlag, 1988), pp. 188–212.Google Scholar
[4]Li, B.-H.. Immersions of manifolds and stable normal bundles. Sci. Sinica Ser. A 30 (1987), 136–147.Google Scholar
[5]Li, B-H.. Normal vector fields of immersions of n-manifolds in (2n−1)-manifolds. Sci. Sinica Ser. A 31 (1988), 31–45.Google Scholar
[6]Li, B.-H. and Peterson, F. P.. On immersions of k-manifolds in (2k−1)-manifolds. Proc. Amer. Math. Soc. 83 (1981), 159–162.Google Scholar
[7]Mahowald, M.. On obstruction theory in orientable fibre bundles. Trans. Amer. Math. Soc. 110 (1964), 315–349.Google Scholar
[8]Olk, C.. Immersionen von Mannigfaltigkeiten in Euklidische Räume. Dissertation, Siegen (1980).Google Scholar