A note on stable homeomorphisms of infinite-dimensional manifolds

Author:
Raymond Y. T. Wong

Journal:
Proc. Amer. Math. Soc. **28** (1971), 271-272

MSC:
Primary 57.55

DOI:
https://doi.org/10.1090/S0002-9939-1971-0271996-8

MathSciNet review:
0271996

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Abstract: In papers by R. D. Anderson and R. Wong, respectively, it is shown that all homeomorphisms of the Hilbert cube onto itself, or of the infinite dimensional separable Hilbert space onto itself, are stable in the sense of Brown-Gluck. These facts can be used to show that all homeomorphisms of onto itself are isotopic to the identity mapping where is either the Hilbert cube or . It follows that some versions of the infinite-dimensional annulus conjecture are true. In this note we give a simple proof of Anderson's result. It follows from Brown-Gluck's technique that for any connected manifold modeled on or , every homeomorphism of onto itself is stable.

**[1]**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**34**#5045. MR**0205212 (34:5045)****[2]**-,*Hilbert space is homeomorphic to the countable infinite product of lines*, Bull. Amer. Math. Soc.**72**(1966), 515-519. MR**32**#8298. MR**0190888 (32:8298)****[3]**-,*On topological infinite deficiency*, Michigan J. Math.**14**(1967), 365-383. MR**35**#4893. MR**0214041 (35:4893)****[4]**M. Brown and H. Gluck,*Stable structures on manifolds*. I.*Homeomorphisms of*, Ann. of Math. (2)**79**(1964), 1-17. MR**28**#1608a. MR**0158383 (28:1608a)****[5]**C. Kuratowski,*Topologie*. Vol. 2, 3rd ed., Monografie Mat., Tom 21, PWN, Warsaw, 1961, p. 32 (7); English transl., Academic Press, New York (to appear). MR**24**#A2958.**[6]**R. Y. T. Wong,*On homeomorphisms of certain infinite dimensional spaces*, Trans. Amer. Math. Soc.**128**(1967), 148-154. MR**35**#4892. MR**0214040 (35:4892)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0271996-8

Keywords:
Homeomorphisms,
stable homeomorphisms,
Hilbert cube

Article copyright:
© Copyright 1971
American Mathematical Society