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‘Line-cobordism’

Published online by Cambridge University Press:  24 October 2008

M. Gate
Affiliation:
Mathematics Department University of Durham, South Road, Durham
G. Whiston
Affiliation:
Mathematics Department University of Durham, South Road, Durham

Extract

The following extension problem is of considerable geometrical interest. Given a compact manifold X = ∂Z the boundary of a compact manifold Z, does the normal bundle of X in Z extend to a line bundle on Z? A related question involving foliations is if X is a compact boundary, is X a leaf of a codimension one foliation on any compact manifold bounded by X? Obviously if it were such a leaf, its normal bundle in the bounded manifold would extend. Reinhart posed both these problems in the language of cobordism theories(1), (2) defining a vector cobordism theory to a deal with questions of the first genre and a foliated-cobordism theory to deal with the second. He was able to obtain complete results in his vector-cobordism theory, whilst the foliated-cobordism theory remains an unsolved problem. Our aim in this article is to extend Reinhart's results on vector-cobordism theory to unoriented line bundles. Whereas in vector-cobordism the Euler number is a cobordism invariant, in the cobordism theory that we shall describe, the modulus of the Euler number becomes a cobordism invariant.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

(1)Reinhart, Bruce L.. Cobordism and the Euler number. Topology 2 (1962), 173177.Google Scholar
(2)Reinhart, Bruce L.. Cobordism and foliations. Ann. Inst. Fourier (Grenoble) 14 (1964), 4952.Google Scholar