On the space of piecewise linear homeomorphisms of a manifold

Geoghegan, Ross; Haver, William E.

doi:10.1090/S0002-9939-1976-0402785-3

On the space of piecewise linear homeomorphisms of a manifold
HTML articles powered by AMS MathViewer

by Ross Geoghegan and William E. Haver PDF

Proc. Amer. Math. Soc. 55 (1976), 145-151 Request permission

Abstract:

Let ${M^n}$ be a compact PL manifold, $n \ne 4$; if $n = 5$, suppose $\partial M$ is empty. Let $H(M)$ be the space of homeomorphisms on $M$ and ${H^{\ast }}(M)$ the elements of $H(M)$ which are isotopic to PL homeomorphisms. It is shown that the space of PL homeomorphisms, $PLH(M)$, has the finite dimensional compact absorption property in ${H^{\ast }}(M)$ and hence that $({H^{\ast }}(M),PLH(M))$ is an $({l_2},l_2^f)$-manifold pair if and only if $H(M)$ is an ${l_2}$-manifold. In particular, if ${M^2}$ is a $2$-manifold, $(H({M^2}),PLH({M^2}))$ is an $({l_2},l_2^f)$-manifold pair.

References

On sigma-compact subsets of infinite-dimensional spaces

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
R. H. Bing, Radial engulfing, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967) Prindle, Weber & Schmidt, Boston, Mass., 1968, pp. 1–18. MR 0238284
R. H. Bing, Locally tame sets are tame, Ann. of Math. (2) 59 (1954), 145–158. MR 61377, DOI 10.2307/1969836
T. A. Chapman, Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. Amer. Math. Soc. 154 (1971), 399–426. MR 283828, DOI 10.1090/S0002-9947-1971-0283828-7
A. V. Černavskiĭ, Local contractibility of the group of homeomorphisms of a manifold. , Dokl. Akad. Nauk SSSR 182 (1968), 510–513 (Russian). MR 0236948
E. H. Connell, Approximating stable homeomorphisms by piecewise linear ones, Ann. of Math. (2) 78 (1963), 326–338. MR 154289, DOI 10.2307/1970346
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
Gordon M. Fisher, On the group of all homeomorphisms of a manifold, Trans. Amer. Math. Soc. 97 (1960), 193–212. MR 117712, DOI 10.1090/S0002-9947-1960-0117712-9

Local contractibility of

for a compact manifold

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, On spaces of homeomorphisms, embeddings, and functions. II. The piecewise linear case, Proc. London Math. Soc. (3) 27 (1973), 463–483. MR 328969, DOI 10.1112/plms/s3-27.3.463
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
Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
James Keesling and David C. Wilson, The group of $PL$-homeomorphisms of a compact $PL$-manifold is an $1^{f}_{2}$-manifold, Trans. Amer. Math. Soc. 193 (1974), 249–256. MR 368046, DOI 10.1090/S0002-9947-1974-0368046-9

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

Über den Begriff Riemannschen Fläche

2

H. Toruńczyk, Absolute retracts as factors of normed linear spaces, Fund. Math. 86 (1974), 53–67. MR 365471, DOI 10.4064/fm-86-1-53-67
James E. West, The ambient homeomorphy of an incomplete subspace of infinite-dimensional Hilbert spaces, Pacific J. Math. 34 (1970), 257–267. MR 277011