Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Structure locale de l'espace des rétractions d'une surface


Author: Robert Cauty
Journal: Trans. Amer. Math. Soc. 323 (1991), 315-334
MSC: Primary 57N20; Secondary 55M15, 57N05, 57S05
DOI: https://doi.org/10.1090/S0002-9947-1991-0994164-8
MathSciNet review: 994164
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Sigma$ be a compact connected $ 2$-manifold, and $ \mathcal{R}(\Sigma )$ the space of retractions of $ \Sigma$. We prove that $ \mathcal{R}(\Sigma )$ is an $ {l^2}$-manifold if the boundary of $ \Sigma$ is not empty, and is the union of an $ {l^2}$-manifold and an isolated point $ {\text{i}}{{\text{d}}_\Sigma }$ if $ \Sigma$ is closed.


References [Enhancements On Off] (What's this?)

  • [1] V. N. Basmanov et A. G. Savchenko, L'espace de Hilbert comme espace des rétractions d'un arc (en russe), Mat. Zametki 42 (1987), 94-100. MR 910032 (88i:54011)
  • [2] K. Borsuk, Concerning the set of retractions, Colloq. Math. 18 (1967), 197-201. MR 0219043 (36:2126)
  • [3] L. Boxer, Retraction spaces and the homotopy metric, Topology Appl. 11 (1980), 17-29. MR 550869 (80k:54024)
  • [4] R. Cauty, Un théorème de sélection et l'espace des rétractions d'une surface, Amer. J. Math. 97 (1975), 282-290. MR 0370587 (51:6814)
  • [5] T. A. Chapman, The space of retractions of a compact Hilbert cube manifold is an A.N.R., Topology Proc. 2 (1977), 409-430. MR 540619 (81e:57014)
  • [6] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
  • [7] E. E. Floyd, On the extension of homeomorphisms on the interior of a two cell, Bull. Amer. Math. Soc. 52 (1946), 654-658. MR 0016675 (8:50e)
  • [8] R. Geoghegan, On spaces of homeomorphisms, embeddings and functions. I, Topology 11 (1972), 159-177. MR 0295281 (45:4349)
  • [9] D. W. Henderson and R. M. Schori, Topological classification of infinite dimensional manifolds by homotopy type, Bull. Amer. Math. Soc. 76 (1970), 121-124. MR 0251749 (40:4976)
  • [10] S. T. Hu, Theory of retracts, Wayne State Univ. Press, Detroit, Mich., 1965. MR 0181977 (31:6202)
  • [11] 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 0301693 (46:849)
  • [12] A. I. Markushevich, Theory of functions of a complex variable. III, Prentice-Hall, Englewood Cliffs, N.J., 1967. MR 0215964 (35:6799)
  • [13] K. Sakai, The space of retractions of a compact $ Q$-manifold is an $ {l^2}$-manifold, Proc. Amer. Math. Soc. 83 (1981), 421-424. MR 624944 (82j:57011)
  • [14] H. Toruńczyk, Absolute retracts as factors of normed linear spaces, Fund. Math. 86 (1974), 53-67. MR 0365471 (51:1723)
  • [15] -, A correction of two papers concerning Hilbert manifolds, Fund. Math. 125 (1985), 89-93. MR 813992 (87m:57017)
  • [16] N. R. Wagner, The space of retractions of the $ 2$-sphere and the annulus, Trans. Amer. Math. Soc. 158 (1971), 319-329. MR 0279763 (43:5484)
  • [17] -, The space of retractions of a $ 2$-manifold, Proc. Amer. Math. Soc. 34 (1972), 609-614. MR 0295282 (45:4350)
  • [18] -, A continuity property with applications to the topology of $ 2$-manifolds, Trans. Amer. Math. Soc. 200 (1974), 369-393. MR 0358781 (50:11240)
  • [19] G. T. Whyburn, Analytic topology, Amer. Math. Soc., Providence, R.I., 1942. MR 0007095 (4:86b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57N20, 55M15, 57N05, 57S05

Retrieve articles in all journals with MSC: 57N20, 55M15, 57N05, 57S05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-0994164-8
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society