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k-Stability of homeomorphisms of Euclidean n-space

Published online by Cambridge University Press:  24 October 2008

W. R. Brakes
W. R. Brakes
Affiliation:
Department of Pure Mathematics, Cambridge University
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Extract

Kirby and Siebenmann (in (7)) proved the annulus conjecture in dimensions five and above, by proving the stable homeomorphism conjecture in those dimensions, that is, by proving that every orientation-preserving homeomorphism of Rn is stable, if n ≥ 5. Although this result apparently lessens the importance of stable homeomorphisms, the concept of stability should still be of use in settling the fate of the annulus conjecture in dimension 4, or for uncovering a simple (i.e. non-surgical) proof in other dimensions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 75 , Issue 2 , March 1974 , pp. 175 - 183
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

(1)Brown, M. and Gluck, H.Stable structures on manifolds, I, II, III. Ann. of Math. 79 (1964), 1.CrossRefGoogle Scholar
(2)Cernavskii, A. V.Homeomorphisms of Euclidean space and topological embeddings of polyhedra in Euclidean spaces. Mat. Sb. 68 (1965), 581. = Amer. Math. Soc. Transl. 78 (1968), 39.Google Scholar
(3)Cernavskii, A. V.Homeomorphisms of Rn are k-stable for k ≤ n– 3. Mat. Sb. 70 (1966), 605. = Amer. Math. Soc. Transl. 78 (1968), 39.Google Scholar
(4)Cernavskii, A. V.The k-stability of homeomorphisms and the union of cells. Soviet Math. Dokl. 9 (1968), 729.Google Scholar
(5)Edwards, R. D. and Kirby, R. C.Deformations of spaces of imbeddings. Ann. of Math. 93 (1971), 63.CrossRefGoogle Scholar
(6)Gluck, H. Embeddings and automorphisms of open manifolds. Topology of Manifolds, edited by Cantrell, J. C.and Edwards, C. H. (Athens Georgia Conf. 1969), Markham, Chicago, 1970.Google Scholar
(7)Kirby, R. C.Stable homeomorphisms and the annulus conjecture. Ann. of Math. 89 (1969), 575.CrossRefGoogle Scholar
(8)Kister, J. M.Microbundles are fiber bundles. Ann. of Math. 80 (1964), 190.CrossRefGoogle Scholar
(9)Stallings, J.On topologically unknotted spheres. Ann. of Math. 77 (1963), 490.CrossRefGoogle Scholar