Stationary isotopies of infinite-dimensional spaces

Wong, Raymond Y. T.

doi:10.1090/S0002-9947-1971-0275476-X

Stationary isotopies of infinite-dimensional spaces
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by Raymond Y. T. Wong

Trans. Amer. Math. Soc. 156 (1971), 131-136

DOI: https://doi.org/10.1090/S0002-9947-1971-0275476-X

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Abstract:

Let X denote the Hilbert cube or any separable infinite-dimensional Fréchet space. It has been shown that any two homeomorphisms f, g of X onto itself is isotopic to each other by means of an invertible-isotopy on X. In this paper we generalize the above results to the extent that if f, g are K-coincident on X (that is, $f(x) = g(x)$ for $x \in K$), then the isotopy can be chosen to be K-stationary provided K is compact and has property-Z in X. The main tool of this paper is the Stable Homeomorphism Extension Theorem which generalizes results of Klee and Anderson.

References

R. D. Anderson, Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 72 (1966), 515–519. MR 190888, DOI 10.1090/S0002-9904-1966-11524-0
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
R. D. Anderson, On topological infinite deficiency, Michigan Math. J. 14 (1967), 365–383. MR 214041, DOI 10.1307/mmj/1028999787
R. D. Anderson, Strongly negligible sets in Fréchet manifolds, Bull. Amer. Math. Soc. 75 (1969), 64–67. MR 238358, DOI 10.1090/S0002-9904-1969-12146-4

Spaces of homeomorphisms of finite graphs

Victor L. Klee Jr., Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc. 74 (1953), 10–43. MR 54850, DOI 10.1090/S0002-9947-1953-0054850-X

Topologie

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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
Raymond Y. T. Wong, A wild Cantor set in the Hilbert cube, Pacific J. Math. 24 (1968), 189–193. MR 221487, DOI 10.2140/pjm.1968.24.189