On homeomorphisms of infinite-dimensional bundles. I
HTML articles powered by AMS MathViewer
- by Raymond Y. T. Wong
- Trans. Amer. Math. Soc. 191 (1974), 245-259
- DOI: https://doi.org/10.1090/S0002-9947-1974-0415625-6
- PDF | Request permission
Abstract:
In this paper we present several aspects of homeomorphism theory in the setting of fibre bundles modeled on separable infinite-dimensional Hilbert (Fréchet) spaces. We study (homotopic) negligibility of subsets, separation of sets, characterization of subsets of infinite-deficiency and extending homeomorphisms; in an essential way they generalize previously known results for manifolds. An important tool is a lemma concerning the lifting of a map to the total space of a bundle whose image misses a certain closed subset presented as obstruction; from this we are able to obtain a result characterizing all subsets of infinite deficiency (for bundles) by their restriction to each fibre. Other results then follow more or less routinely by employing the rather standard methods of infinite-dimensional topology.References
- R. D. Anderson, Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 72 (1966), 515–519. MR 190888, DOI 10.1090/S0002-9904-1966-11524-0
- R. D. Anderson, On topological infinite deficiency, Michigan Math. J. 14 (1967), 365–383. MR 214041, DOI 10.1307/mmj/1028999787
- R. D. Anderson, Topological properties of the Hilbert cube and the infinite product of open intervals, Trans. Amer. Math. Soc. 126 (1967), 200–216. MR 205212, DOI 10.1090/S0002-9947-1967-0205212-3
- R. D. Anderson and R. H. Bing, A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 74 (1968), 771–792. MR 230284, DOI 10.1090/S0002-9904-1968-12044-0
- R. D. Anderson, David W. Henderson, and James E. West, Negligible subsets of infinite-dimensional manifolds, Compositio Math. 21 (1969), 143–150. MR 246326
- R. D. Anderson and John D. McCharen, On extending homeomorphisms to Fréchet manifolds, Proc. Amer. Math. Soc. 25 (1970), 283–289. MR 258064, DOI 10.1090/S0002-9939-1970-0258064-5
- R. D. Anderson and R. Schori, Factors of infinite-dimensional manifolds, Trans. Amer. Math. Soc. 142 (1969), 315–330. MR 246327, DOI 10.1090/S0002-9947-1969-0246327-5 W. Barit, Some properties of certain subsets of infinite-dimensional spaces, Dissertation, Lousiana State University, 1971.
- Czesław Bessaga and Victor Klee, Every non-normable Frechet space is homeomorphic with all of its closed convex bodies, Math. Ann. 163 (1966), 161–166. MR 201949, DOI 10.1007/BF02052848
- T. A. Chapman, Four classes of separable metric infinite-dimensional manifolds, Bull. Amer. Math. Soc. 76 (1970), 399–403. MR 253375, DOI 10.1090/S0002-9904-1970-12490-9
- T. A. Chapman, Infinite deficiency in Fréchet manifolds, Trans. Amer. Math. Soc. 148 (1970), 137–146. MR 256418, DOI 10.1090/S0002-9947-1970-0256418-9
- William H. Cutler, Negligible subsets of infinite-dimensional Fréchet manifolds, Proc. Amer. Math. Soc. 23 (1969), 668–675. MR 248883, DOI 10.1090/S0002-9939-1969-0248883-5
- James Eells Jr. and Nicolaas H. Kuiper, Homotopy negligible subsets, Compositio Math. 21 (1969), 155–161. MR 253331
- David W. Henderson, Infinite-dimensional manifolds are open subsets of Hilbert space, Bull. Amer. Math. Soc. 75 (1969), 759–762. MR 247634, DOI 10.1090/S0002-9904-1969-12276-7
- Dale Husemoller, Fibre bundles, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR 0229247, DOI 10.1007/978-1-4757-4008-0
- Victor L. Klee Jr., Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc. 74 (1953), 10–43. MR 54850, DOI 10.1090/S0002-9947-1953-0054850-X
- Richard S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1–16. MR 189028, DOI 10.1016/0040-9383(66)90002-4
- Peter L. Renz, The contractibility of the homeomorphism group of some product spaces by Wong’s method, Math. Scand. 28 (1971), 182–188. MR 305426, DOI 10.7146/math.scand.a-11014
- James E. West, Fixed-point sets of transformation groups on infinite-product spaces, Proc. Amer. Math. Soc. 21 (1969), 575–582. MR 239588, DOI 10.1090/S0002-9939-1969-0239588-5
- Raymond Y. T. Wong, On homeomorphisms of certain infinite dimensional spaces, Trans. Amer. Math. Soc. 128 (1967), 148–154. MR 214040, DOI 10.1090/S0002-9947-1967-0214040-4
- Raymond Y. T. Wong, Stationary isotopies of infinite-dimensional spaces, Trans. Amer. Math. Soc. 156 (1971), 131–136. MR 275476, DOI 10.1090/S0002-9947-1971-0275476-X
- Raymond Y. T. Wong, Homotopy negligible subsets of bundles, Compositio Math. 24 (1972), 119–128. MR 307269
- Raymond Y. T. Wong, On homeomorphisms of infinite-dimensional bundles. I, Trans. Amer. Math. Soc. 191 (1974), 245–259. MR 415625, DOI 10.1090/S0002-9947-1974-0415625-6
- Raymond Y. T. Wong, On homeomorphisms of infinite-dimensional bundles. I, Trans. Amer. Math. Soc. 191 (1974), 245–259. MR 415625, DOI 10.1090/S0002-9947-1974-0415625-6
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 191 (1974), 245-259
- MSC: Primary 57A20; Secondary 58B05
- DOI: https://doi.org/10.1090/S0002-9947-1974-0415625-6
- MathSciNet review: 0415625