## Homeomorphism groups of some direct limit spaces

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- by Margie Hale
- Proc. Amer. Math. Soc.
**85**(1982), 661-665 - DOI: https://doi.org/10.1090/S0002-9939-1982-0660625-4
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## Abstract:

Let $F$ be either of the spaces ${R^\infty } = {R^n}$ or ${Q^\infty } = {Q^n}$ where $R$ denotes the reals and $Q$ the Hilbert cube. Let $\mathcal {H}(M)$ be the homeomorphism group of a connected $F$-manifold $M$ with the compact-open topology. We prove that $\mathcal {H}(M)$ is separable, LindelΓΆf, paracompact, non-first-countable, and not a $k$-space.## References

- R. D. Anderson,
*Hilbert space is homeomorphic to the countable infinite product of lines*, Bull. Amer. Math. Soc.**72**(1966), 515β519. MR**190888**, DOI 10.1090/S0002-9904-1966-11524-0 - James Dugundji,
*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606** - Steve Ferry,
*The homeomorphism group of a compact Hilbert cube manifold is an $\textrm {ANR}$*, Ann. of Math. (2)**106**(1977), no.Β 1, 101β119. MR**461536**, DOI 10.2307/1971161 - Margie Hale,
*A factoring technique for homeomorphism groups*, Topology Proc.**6**(1981), no.Β 2, 299β309 (1982). MR**672461** - Vagn Lundsgaard Hansen,
*Some theorems on direct limits of expanding sequences of manifolds*, Math. Scand.**29**(1971), 5β36. MR**319206**, DOI 10.7146/math.scand.a-11031 - Richard E. Heisey,
*Contracting spaces of maps on the countable direct limit of a space*, Trans. Amer. Math. Soc.**193**(1974), 389β411. MR**367908**, DOI 10.1090/S0002-9947-1974-0367908-6 - Richard E. Heisey,
*Manifolds modelled on $R^{\infty }$ or bounded weak-* topologies*, Trans. Amer. Math. Soc.**206**(1975), 295β312. MR**397768**, DOI 10.1090/S0002-9947-1975-0397768-X
β, - Vo Thanh Liem,
*An $\alpha$-approximation theorem for $Q^{\infty }$-manifolds*, Topology Appl.**12**(1981), no.Β 3, 289β304. MR**623737**, DOI 10.1016/0166-8641(81)90007-9 - Vo Thanh Liem,
*An unknotting theorem in $Q^{\infty }$-manifolds*, Proc. Amer. Math. Soc.**82**(1981), no.Β 1, 125β132. MR**603615**, DOI 10.1090/S0002-9939-1981-0603615-9
β, - Peter L. Renz,
*The contractibility of the homeomorphism group of some product spaces by Wongβs method*, Math. Scand.**28**(1971), 182β188. MR**305426**, DOI 10.7146/math.scand.a-11014

*Manifolds modeled on the direct limit of Hilbert cubes*, Proc. Conf. Geometric Topology, Univ. of Georgia, 1977. β,

*Stability, classification, open embeddings, and triangulation of*${R^\infty }$

*-manifolds*, Proc. Internat. Conf. Geometric Topology, Polish Scientific Publishers, Warsaw, 1980, pp. 193-196.

*An*$\alpha$

*-approximation theorem for*${R^\infty }$

*-manifolds*, preprint.

## Bibliographic Information

- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**85**(1982), 661-665 - MSC: Primary 57S05; Secondary 54H15, 57N20, 58B05
- DOI: https://doi.org/10.1090/S0002-9939-1982-0660625-4
- MathSciNet review: 660625