Characterizing the topology of infinite-dimensional -compact manifolds

Author:
Jerzy Mogilski

Journal:
Proc. Amer. Math. Soc. **92** (1984), 111-118

MSC:
Primary 57N20; Secondary 54C55

MathSciNet review:
749902

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A metric space , which is a countable union of finite-dimensional compacta, is a manifold modelled on the space all but finitely many iff is an ANR and the following condition holds: given , a pair of finite-dimensional compacta and a map such that is an embedding, there is an embedding such that and for all . An analogous condition characterizes manifolds modelled on the space .

**[1]**R. D. Anderson,*On sigma-compact subsets of finite-dimensional spaces*, preprint.**[2]**R. D. Anderson, D. W. Curtis, and J. van Mill,*A fake topological Hilbert space*, Trans. Amer. Math. Soc.**272**(1982), no. 1, 311–321. MR**656491**, 10.1090/S0002-9947-1982-0656491-8**[3]**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**0273347****[4]**-,*Selected topics in infinite-dimensional topology*, PWN, Warsaw, 1975.**[5]**D. W. Curtis,*Hyperspaces of finite subsets as boundary sets*, Topology Appl.**22**(1986), no. 1, 97–107. MR**831185**, 10.1016/0166-8641(86)90081-7**[6]**D. W. Curtis, T. Dobrowolski and J. Mogilski,*Some applications of the topological characterizations of the sigma-compact spaces*, and , Trans. Amer. Math. Soc. (to appear).**[7]**T. Dobrowolski and J. Mogilski,*Sigma-compact locally convex metric linear spaces universal for compacta are homeomorphic*, Proc. Amer. Math. Soc.**85**(1982), no. 4, 653–658. MR**660623**, 10.1090/S0002-9939-1982-0660623-0**[8]**Ross Geoghegan,*Manifolds of piecewise linear maps and a related normed linear space.*, Bull. Amer. Math. Soc.**77**(1971), 629–632. MR**0277010**, 10.1090/S0002-9904-1971-12782-9**[9]**Ross Geoghegan (ed.),*Open problems in infinite-dimensional topology*, The Proceedings of the 1979 Topology Conference (Ohio Univ., Athens, Ohio, 1979), 1979, pp. 287–338 (1980). MR**583711****[10]**Ross Geoghegan and William E. Haver,*On the space of piecewise linear homeomorphisms of a manifold*, Proc. Amer. Math. Soc.**55**(1976), no. 1, 145–151. MR**0402785**, 10.1090/S0002-9939-1976-0402785-3**[11]**James P. Henderson and John J. Walsh,*Examples of cell-like decompositions of the infinite-dimensional manifolds 𝜎 and Σ*, Topology Appl.**16**(1983), no. 2, 143–154. MR**712860**, 10.1016/0166-8641(83)90014-7**[12]**Jerzy Mogilski,*CE-decomposition of 𝑙₂-manifolds*, Bull. Acad. Polon. Sci. Sér. Sci. Math.**27**(1979), no. 3-4, 309–314 (English, with Russian summary). MR**552055****[13]**J. Mogilski and H. Roslaniec,*Cell-like decompositions of ANR's*, Topology Appl. (submitted).**[14]**H. Torunczyk,*Skeletons and absorbing sets in complete meric spaces*, Doctoral thesis, Institute of Mathematics of the Polish Academic of Sciences, 1970.**[15]**H. Toruńczyk,*Concerning locally homotopy negligible sets and characterization of 𝑙₂-manifolds*, Fund. Math.**101**(1978), no. 2, 93–110. MR**518344****[16]**-,*An introduction to infinite-dimensional topology*, in preparation.**[17]**H. Toruńczyk,*Absolute retracts as factors of normed linear spaces*, Fund. Math.**86**(1974), 53–67. MR**0365471****[18]**H. Toruńczyk,*On 𝐶𝐸-images of the Hilbert cube and characterization of 𝑄-manifolds*, Fund. Math.**106**(1980), no. 1, 31–40. MR**585543****[19]**H. Toruńczyk,*Characterizing Hilbert space topology*, Fund. Math.**111**(1981), no. 3, 247–262. MR**611763****[20]**James E. West,*The ambient homeomorphy of an incomplete subspace of infinite-dimensional Hilbert spaces*, Pacific J. Math.**34**(1970), 257–267. MR**0277011****[21]**Ph. Bowers, M. Bestvina, J. Mogilski and J. Walsh,*Characterizing Hilbert space manifolds revisited*, in preparation.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
57N20,
54C55

Retrieve articles in all journals with MSC: 57N20, 54C55

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1984-0749902-8

Keywords:
Sigma-compact metric ANR's,
the strong universality property for compacta,
infinite-dimensional sigma-compact manifolds,
-sets,
near-homeomorphisms

Article copyright:
© Copyright 1984
American Mathematical Society