Stable homeomorphisms on infinite-dimensional normed linear spaces.

Curtis, D. W.; McCoy, R. A.

doi:10.1090/S0002-9939-1971-0283831-2

Stable homeomorphisms on infinite-dimensional normed linear spaces.
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by D. W. Curtis and R. A. McCoy PDF

Proc. Amer. Math. Soc. 28 (1971), 496-500 Request permission

Abstract:

R. Y. T. Wong has recently shown that all homeomorphisms on a connected manifold modeled on infinite-dimensional separable Hilbert space are stable. In this paper we establish the stability of all homeomorphisms on a normed linear space $E$ such that $E$ is homeomorphic to the countable infinite product of copies of itself. The relationship between stability of homeomorphisms and a strong annulus conjecture is demonstrated and used to show that stability of all homeomorphisms on a normed linear space $E$ implies stability of all homeomorphisms on a connected manifold modeled on $E$, and that in such a manifold collared $E$-cells are tame.

References

R. D. Anderson, Topological properties of the Hilbert cube and the infinite product of open intervals, Trans. Amer. Math. Soc. 126 (1967), 200–216. MR 205212, DOI 10.1090/S0002-9947-1967-0205212-3
C. Bessaga and M. I. Kadec, On topological classification of non-separable Banach spaces, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967) Ann. of Math. Studies, No. 69, Princeton Univ. Press, Princeton, N. J., 1972, pp. 15–24. MR 0417765
Marston Morse, A reduction of the Schoenflies extension problem, Bull. Amer. Math. Soc. 66 (1960), 113–115. MR 117694, DOI 10.1090/S0002-9904-1960-10420-X
Morton Brown and Herman Gluck, Stable structures on manifolds. I. Homeomorphisms of $S^{n}$, Ann. of Math. (2) 79 (1964), 1–17. MR 158383, DOI 10.2307/1970481

Deficiency in $F$-manifolds

J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353–367. MR 44116, DOI 10.2140/pjm.1951.1.353
V. L. Klee Jr., Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30–45. MR 69388, DOI 10.1090/S0002-9947-1955-0069388-5
V. L. Klee Jr., A note on topological properties of normed linear spaces, Proc. Amer. Math. Soc. 7 (1956), 673–674. MR 78661, DOI 10.1090/S0002-9939-1956-0078661-2
R. A. McCoy, Annulus conjecture and stability of homeomorphisms in infinite-dimensional normed linear spaces, Proc. Amer. Math. Soc. 24 (1970), 272–277. MR 256419, DOI 10.1090/S0002-9939-1970-0256419-6
D. E. Sanderson, An infinite-dimensional Schoenflies theorem, Trans. Amer. Math. Soc. 148 (1970), 33–39. MR 259957, DOI 10.1090/S0002-9947-1970-0259957-X
James V. Whittaker, Some normal subgroups of homomorphisms, Trans. Amer. Math. Soc. 123 (1966), 88–98. MR 192482, DOI 10.1090/S0002-9947-1966-0192482-2
Raymond Y. T. Wong, On homeomorphisms of certain infinite dimensional spaces, Trans. Amer. Math. Soc. 128 (1967), 148–154. MR 214040, DOI 10.1090/S0002-9947-1967-0214040-4

On stable homeomorphisms of infinite-dimensional manifolds