Homeomorphism groups of some direct limit spaces

Hale, Margie

doi:10.1090/S0002-9939-1982-0660625-4

Homeomorphism groups of some direct limit spaces
HTML articles powered by AMS MathViewer

by Margie Hale

Proc. Amer. Math. Soc. 85 (1982), 661-665

DOI: https://doi.org/10.1090/S0002-9939-1982-0660625-4

PDF | Request permission

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, Lindelöf, 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), 515–519. 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. 1, 101–119. MR 461536, DOI 10.2307/1971161
Margie Hale, A factoring technique for homeomorphism groups, Topology Proc. 6 (1981), no. 2, 299–309 (1982). MR 672461
Vagn Lundsgaard Hansen, Some theorems on direct limits of expanding sequences of manifolds, Math. Scand. 29 (1971), 5–36. 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), 389–411. 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), 295–312. MR 397768, DOI 10.1090/S0002-9947-1975-0397768-X

Manifolds modeled on the direct limit of Hilbert cubes

Stability, classification, open embeddings, and triangulation of

-manifolds

Vo Thanh Liem, An $\alpha$-approximation theorem for $Q^{\infty }$-manifolds, Topology Appl. 12 (1981), no. 3, 289–304. 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. 1, 125–132. MR 603615, DOI 10.1090/S0002-9939-1981-0603615-9

An

-approximation theorem for

-manifolds

Peter L. Renz, The contractibility of the homeomorphism group of some product spaces by Wong’s method, Math. Scand. 28 (1971), 182–188. MR 305426, DOI 10.7146/math.scand.a-11014