Recognizing $\sigma$-manifolds
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- by James P. Henderson PDF
- Proc. Amer. Math. Soc. 94 (1985), 721-727 Request permission
Abstract:
Denote by $\sigma$ the subspace of the Hilbert cube consisting of $\{ (x_i): x_i = 0$ for all but finitely many $i$}. Then following characterization of manifolds modeled on $\sigma$ is proven and applied to cell-like, upper semicontinuous decompositions of $\sigma$-manifolds. An ANR $X$ is a $\sigma$-manifold if and only if (a) $X$ is the countable union of finite-dimensional compacta, (b) each compact subset of $X$ is a strong $Z$-set, and (c) for each positive integer $k$, every mapping $f:{R^k} \to X$ can be arbitrarily closely approximated by an injection.References
- Fredric D. Ancel, The role of countable dimensionality in the theory of cell-like relations, Trans. Amer. Math. Soc. 287 (1985), no. 1, 1–40. MR 766204, DOI 10.1090/S0002-9947-1985-0766204-X
- 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
- J. W. Cannon, The recognition problem: what is a topological manifold?, Bull. Amer. Math. Soc. 84 (1978), no. 5, 832–866. MR 494113, DOI 10.1090/S0002-9904-1978-14527-3
- J. W. Cannon, Shrinking cell-like decompositions of manifolds. Codimension three, Ann. of Math. (2) 110 (1979), no. 1, 83–112. MR 541330, DOI 10.2307/1971245
- 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
- Robert J. Daverman, Products of cell-like decompositions, Topology Appl. 11 (1980), no. 2, 121–139. MR 572368, DOI 10.1016/0166-8641(80)90002-4 R. D. Edwards, Approximating certain cell-like maps by homeomorphisms, Preprint. See Notices Amer. Math. Soc. 24 (1977), A-649, #751-G5.
- James P. Henderson and John J. Walsh, Examples of cell-like decompositions of the infinite-dimensional manifolds $\sigma$ and $\Sigma$, Topology Appl. 16 (1983), no. 2, 143–154. MR 712860, DOI 10.1016/0166-8641(83)90014-7
- R. C. Lacher, Cell-like mappings and their generalizations, Bull. Amer. Math. Soc. 83 (1977), no. 4, 495–552. MR 645403, DOI 10.1090/S0002-9904-1977-14321-8
- Jerzy Mogilski, Characterizing the topology of infinite-dimensional $\sigma$-compact manifolds, Proc. Amer. Math. Soc. 92 (1984), no. 1, 111–118. MR 749902, DOI 10.1090/S0002-9939-1984-0749902-8
- Frank Quinn, Ends of maps. I, Ann. of Math. (2) 110 (1979), no. 2, 275–331. MR 549490, DOI 10.2307/1971262
- H. Toruńczyk, Skeletonized sets in complete metric spaces and homeomorphisms of the Hilbert cube, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 119–126 (English, with Russian summary). MR 264602
- H. Toruńczyk, On $\textrm {CE}$-images of the Hilbert cube and characterization of $Q$-manifolds, Fund. Math. 106 (1980), no. 1, 31–40. MR 585543, DOI 10.4064/fm-106-1-31-40 J. J. Walsh, Detecting finite and infinite dimensional manifolds, Address to 811th meeting of Amer. Math. Soc., April 13, 1984, Richmond, Va. See Abstracts Amer. Math. Soc. 5 (1984), 182, #811-57-01.
- James E. West, The ambient homeomorphy of an incomplete subspace of infinite-dimensional Hilbert spaces, Pacific J. Math. 34 (1970), 257–267. MR 277011, DOI 10.2140/pjm.1970.34.257
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 94 (1985), 721-727
- MSC: Primary 57N20; Secondary 54B15, 54C99, 58B05
- DOI: https://doi.org/10.1090/S0002-9939-1985-0792291-4
- MathSciNet review: 792291