The group of 𝑃𝐿-homeomorphisms of a compact 𝑃𝐿-manifold is an 1^{𝑓}₂-manifold

Keesling, James; Wilson, David C.

doi:10.1090/S0002-9947-1974-0368046-9

The group of $PL$-homeomorphisms of a compact $PL$-manifold is an $1^{f}_{2}$-manifold
HTML articles powered by AMS MathViewer

by James Keesling and David C. Wilson

Trans. Amer. Math. Soc. 193 (1974), 249-256

DOI: https://doi.org/10.1090/S0002-9947-1974-0368046-9

PDF | Request permission

Abstract:

In this paper it is shown that if $M$ is a compact PL-manifold and ${H_{PL}}(M)$ is the group of PL-homeomorphisms of $M$ onto itself, then ${H_{PL}}(M)$ is an $l_2^f$-manifold. Here ${l_2}$ is the Hilbert space of all real-valued square-summable sequences and $l_2^f = \{ ({x_i}) \in {l_2}:{x_i} = 0$ for almost all $i$.

References

C. Bessaga and A. Pełczyński, The estimated extension theorem, homogeneous collections and skeletons, and their applications to the topological classification of linear metric spaces and convex sets, Fund. Math. 69 (1970), 153–190. MR 273347, DOI 10.4064/fm-69-2-153-190
A. V. Černavskiĭ, Local contractibility of the group of homeomorphisms of a manifold. , Dokl. Akad. Nauk SSSR 182 (1968), 510–513 (Russian). MR 0236948
A. V. Černavskiĭ, Local contractibility of the group of homeomorphisms of a manifold. , Mat. Sb. (N.S.) 79 (121) (1969), 307–356 (Russian). MR 0259925
Robert D. Edwards and Robion C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63–88. MR 283802, DOI 10.2307/1970753
Ross Geoghegan, On spaces of homeomorphisms, embeddings and functions. I, Topology 11 (1972), 159–177. MR 295281, DOI 10.1016/0040-9383(72)90004-3
Ross Geoghegan, Manifolds of piecewise linear maps and a related normed linear space, Bull. Amer. Math. Soc. 77 (1971), 629–632. MR 277010, DOI 10.1090/S0002-9904-1971-12782-9

On spaces of homeomorphisms, imbeddings and functions

The piecewise linear case

William E. Haver, Locally contractible spaces that are absolute neighborhood retracts, Proc. Amer. Math. Soc. 40 (1973), 280–284. MR 331311, DOI 10.1090/S0002-9939-1973-0331311-X
David W. Henderson and James E. West, Triangulated infinite-dimensional manifolds, Bull. Amer. Math. Soc. 76 (1970), 655–660. MR 258067, DOI 10.1090/S0002-9904-1970-12478-8
James Keesling, Using flows to construct Hilbert space factors of function spaces, Trans. Amer. Math. Soc. 161 (1971), 1–24. MR 283751, DOI 10.1090/S0002-9947-1971-0283751-8
James Keesling, The group of homeomorphisms of a solenoid, Trans. Amer. Math. Soc. 172 (1972), 119–131. MR 315735, DOI 10.1090/S0002-9947-1972-0315735-6

Lectures on triangulations of manifolds

R. Luke and W. K. Mason, The space of homeomorphisms on a compact two-manifold is an absolute neighborhood retract, Trans. Amer. Math. Soc. 164 (1972), 275–285. MR 301693, DOI 10.1090/S0002-9947-1972-0301693-7
W. K. Mason, The space of all self-homeomorphisms of a two-cell which fix the cell’s boundary is an absolute retract, Trans. Amer. Math. Soc. 161 (1971), 185–205. MR 286067, DOI 10.1090/S0002-9947-1971-0286067-9
Marston Morse, A special parameterization of curves, Bull. Amer. Math. Soc. 42 (1936), no. 12, 915–922. MR 1563464, DOI 10.1090/S0002-9904-1936-06466-9
T. Benny Rushing, Topological embeddings, Pure and Applied Mathematics, Vol. 52, Academic Press, New York-London, 1973. MR 0348752

Compact absolute retracts as factors of the Hilbert space

Absolute retracts as factors of normed linear spaces