On $\textbf {R}^ \infty \;(Q^ \infty )$-manifold bundles over CW complexes
HTML articles powered by AMS MathViewer
- by Vo Thanh Liem PDF
- Trans. Amer. Math. Soc. 297 (1986), 563-585 Request permission
Abstract:
Let $\Lambda \in \mathcal {C}\mathcal {W}(\mathcal {C}) \cup \mathcal {C}\mathcal {W}(\mathcal {M})$ be a pseudo CW complex generated either by Hausdorff compact spaces or by metric spaces. In the theory of manifolds modeled on ${R^\infty }$ or ${Q^\infty }$, we will prove the $\Lambda$-fiber-preserving versions of the following: Equivalences among the notions of $D$-sets, ${D^{\ast }}$-sets and infinite deficient sets; relative stability theorems; relative deformation of homotopy equivalences to homeomorphisms; strong unknotting theorem for $D$-embeddings; and $\alpha$-approximation theorems.References
- Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
- T. A. Chapman, Lectures on Hilbert cube manifolds, Regional Conference Series in Mathematics, No. 28, American Mathematical Society, Providence, R.I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975. MR 0423357, DOI 10.1090/cbms/028
- T. A. Chapman and Steve Ferry, Hurewicz fiber maps with ANR fibers, Topology 16 (1977), no.Β 2, 131β143. MR 448356, DOI 10.1016/0040-9383(77)90011-8
- James Dugundji, Topology, Allyn and Bacon Series in Advanced Mathematics, Allyn and Bacon, Inc., Boston, Mass.-London-Sydney, 1978. Reprinting of the 1966 original. MR 0478089
- Ryszard Engelking, Topologia ogΓ³lna, PaΕstwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- Steve Ferry, The homeomorphism group of a compact Hilbert cube manifold is an $\textrm {ANR}$, Ann. of Math. (2) 106 (1977), no.Β 1, 101β119. MR 461536, DOI 10.2307/1971161
- Brayton Gray, Homotopy theory, Pure and Applied Mathematics, Vol. 64, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. An introduction to algebraic topology. MR 0402714
- Marvin J. Greenberg, Lectures on algebraic topology, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0215295
- Vagn Lundsgaard Hansen, Some theorems on direct limits of expanding sequences of manifolds, Math. Scand. 29 (1971), 5β36. MR 319206, DOI 10.7146/math.scand.a-11031
- Richard E. Heisey, Stability, open embeddings, and classification of bounded weak-* manifolds, General Topology and Appl. 6 (1976), no.Β 3, 327β338. MR 423394, DOI 10.1016/0016-660X(76)90023-4 β, Stability, classification on open embeddings, and triangulation of ${R^\infty }$-manifolds (K. Borsuk and A. Kirkor, eds.), Proc. Internat. Conf. on Geometric Topology, PWN, Warsaw, 1980, pp. 193-196.
- Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
- D. M. Hyman, A category slightly larger than the metric and $\textrm {CW}$-categories, Michigan Math. J. 15 (1968), 193β214. MR 229198, DOI 10.1307/mmj/1028999974
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
- V. L. Klee Jr., Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30β45. MR 69388, DOI 10.1090/S0002-9947-1955-0069388-5
- Vo Thanh Liem, An $\alpha$-approximation theorem for $Q^{\infty }$-manifolds, Topology Appl. 12 (1981), no.Β 3, 289β304. MR 623737, DOI 10.1016/0166-8641(81)90007-9 β, An $\alpha$-approximation theorem for ${R^\infty }$-manifolds, Rocky Mountain J. Math. (to appear).
- Vo Thanh Liem, On infinite deficiency in $\textbf {R}^\infty$-manifolds, Trans. Amer. Math. Soc. 288 (1985), no.Β 1, 205β226. MR 773057, DOI 10.1090/S0002-9947-1985-0773057-2
- Vo Thanh Liem, An unknotting theorem in $Q^{\infty }$-manifolds, Proc. Amer. Math. Soc. 82 (1981), no.Β 1, 125β132. MR 603615, DOI 10.1090/S0002-9939-1981-0603615-9 β, A local strong $U{V^\infty }$-property of the homeomorphism groups of ${R^\infty }$ (or ${Q^\infty }$)-manifolds, Proc. Amer. Math. Soc. (to appear).
- Ernest Michael, Local properties of topological spaces, Duke Math. J. 21 (1954), 163β171. MR 62424
- Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361β382. MR 77107, DOI 10.2307/1969615
- Katsuro Sakai, On $\textbf {R}^{\infty }$-manifolds and $Q^{\infty }$-manifolds, Topology Appl. 18 (1984), no.Β 1, 69β79. MR 759140, DOI 10.1016/0166-8641(84)90032-4
- Katsuro Sakai, On $\textbf {R}^{\infty }$-manifolds and $Q^{\infty }$-manifolds. II. Infinite deficiency, Tsukuba J. Math. 8 (1984), no.Β 1, 101β118. MR 747449, DOI 10.21099/tkbjm/1496159948 H. TorunΔzyk and J. E. West, Fibrations and bundles with Hilbert-cube-manifold fibers, preprint.
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 297 (1986), 563-585
- MSC: Primary 57N20; Secondary 57N35
- DOI: https://doi.org/10.1090/S0002-9947-1986-0854085-6
- MathSciNet review: 854085