Stable homeomorphisms on infinite-dimensional normed linear spaces.

Authors:
D. W. Curtis and R. A. McCoy

Journal:
Proc. Amer. Math. Soc. **28** (1971), 496-500

MSC:
Primary 57.55; Secondary 46.00

MathSciNet review:
0283831

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 such that 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 implies stability of all homeomorphisms on a connected manifold modeled on , and that in such a manifold collared -cells are tame.

**[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**0205212**, 10.1090/S0002-9947-1967-0205212-3**[2]**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) Princeton Univ. Press, Princeton, N. J., 1972, pp. 15–24. Ann. of Math. Studies, No. 69. MR**0417765****[3]**Morton Brown,*A proof of the generalized Schoenflies theorem*, Bull. Amer. Math. Soc.**66**(1960), 74–76. MR**0117695**, 10.1090/S0002-9904-1960-10400-4**[4]**Morton Brown and Herman Gluck,*Stable structures on manifolds. I. Homeomorphisms of 𝑆ⁿ*, Ann. of Math. (2)**79**(1964), 1–17. MR**0158383****[5]**W. H. Cutler,*Deficiency in -manifolds*(submitted).**[6]**J. Dugundji,*An extension of Tietze’s theorem*, Pacific J. Math.**1**(1951), 353–367. MR**0044116****[7]**V. L. Klee Jr.,*Some topological properties of convex sets*, Trans. Amer. Math. Soc.**78**(1955), 30–45. MR**0069388**, 10.1090/S0002-9947-1955-0069388-5**[8]**V. L. Klee Jr.,*A note on topological properties of normed linear spaces*, Proc. Amer. Math. Soc.**7**(1956), 673–674. MR**0078661**, 10.1090/S0002-9939-1956-0078661-2**[9]**R. A. McCoy,*Annulus conjecture and stability of homeomorphisms in infinite-dimensional normed linear spaces*, Proc. Amer. Math. Soc.**24**(1970), 272–277. MR**0256419**, 10.1090/S0002-9939-1970-0256419-6**[10]**D. E. Sanderson,*An infinite-dimensional Schoenflies theorem*, Trans. Amer. Math. Soc.**148**(1970), 33–39. MR**0259957**, 10.1090/S0002-9947-1970-0259957-X**[11]**James V. Whittaker,*Some normal subgroups of homomorphisms*, Trans. Amer. Math. Soc.**123**(1966), 88–98. MR**0192482**, 10.1090/S0002-9947-1966-0192482-2**[12]**Raymond Y. T. Wong,*On homeomorphisms of certain infinite dimensional spaces*, Trans. Amer. Math. Soc.**128**(1967), 148–154. MR**0214040**, 10.1090/S0002-9947-1967-0214040-4**[13]**-,*On stable homeomorphisms of infinite-dimensional manifolds*(submitted).

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
57.55,
46.00

Retrieve articles in all journals with MSC: 57.55, 46.00

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1971-0283831-2

Keywords:
Infinite-dimensional normed linear spaces,
connected infinite-dimensional manifolds,
stable homeomorphisms,
collared cells,
annulus conjecture

Article copyright:
© Copyright 1971
American Mathematical Society