Structure locale de l’espace des rétractions d’une surface
HTML articles powered by AMS MathViewer
- by Robert Cauty PDF
- Trans. Amer. Math. Soc. 323 (1991), 315-334 Request permission
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
- V. N. Basmanov and A. G. Savchenko, Hilbert space as the space of retractions of a segment, Mat. Zametki 42 (1987), no. 1, 94–100, 168 (Russian). MR 910032
- Karol Borsuk, Concerning the set of retractions, Colloq. Math. 18 (1967), 197–201. MR 219043, DOI 10.4064/cm-18-1-197-201
- Laurence Boxer, Retraction spaces and the homotopy metric, Topology Appl. 11 (1980), no. 1, 17–29. MR 550869, DOI 10.1016/0166-8641(80)90013-9
- Robert Cauty, Un théorème de sélection et l’espace des rétractions d’une surface, Amer. J. Math. 97 (1975), 282–290 (French). MR 370587, DOI 10.2307/2373672
- T. A. Chapman, The space of retractions of a compact Hilbert cube manifold is an ANR, Topology Proc. 2 (1977), no. 2, 409–430 (1978). MR 540619
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
- E. E. Floyd, On the extension of homeomorphisms on the interior of a two cell, Bull. Amer. Math. Soc. 52 (1946), 654–658. MR 16675, DOI 10.1090/S0002-9904-1946-08617-6
- Ross Geoghegan, On spaces of homeomorphisms, embeddings and functions. I, Topology 11 (1972), 159–177. MR 295281, DOI 10.1016/0040-9383(72)90004-3
- David W. Henderson and R. Schori, Topological classification of infinite dimensional manifolds by homotopy type, Bull. Amer. Math. Soc. 76 (1970), 121–124. MR 251749, DOI 10.1090/S0002-9904-1970-12392-8
- Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
- 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 301693, DOI 10.1090/S0002-9947-1972-0301693-7
- A. I. Markushevich, Theory of functions of a complex variable. Vol. III, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. Revised English edition, translated and edited by Richard A. Silverman. MR 0215964
- Katsuro Sakai, The space of retractions of a compact $Q$-manifold is an $l^{2}$-manifold, Proc. Amer. Math. Soc. 83 (1981), no. 2, 421–424. MR 624944, DOI 10.1090/S0002-9939-1981-0624944-9
- H. Toruńczyk, Absolute retracts as factors of normed linear spaces, Fund. Math. 86 (1974), 53–67. MR 365471, DOI 10.4064/fm-86-1-53-67
- H. Toruńczyk, A correction of two papers concerning Hilbert manifolds: “Concerning locally homotopy negligible sets and characterization of $l_2$-manifolds” [Fund. Math. 101 (1978), no. 2, 93–110; MR0518344 (80g:57019)] and “Characterizing Hilbert space topology” [ibid. 111 (1981), no. 3, 247–262; MR0611763 (82i:57016)], Fund. Math. 125 (1985), no. 1, 89–93. MR 813992, DOI 10.4064/fm-125-1-89-93
- Neal R. Wagner, The space of retractions of the $2$-sphere and the annulus, Trans. Amer. Math. Soc. 158 (1971), 319–329. MR 279763, DOI 10.1090/S0002-9947-1971-0279763-0
- Neal R. Wagner, The space of retractions of a $2$-manifold, Proc. Amer. Math. Soc. 34 (1972), 609–614. MR 295282, DOI 10.1090/S0002-9939-1972-0295282-6
- Neal R. Wagner, A continuity property with applications to the topology of $2$-manifolds, Trans. Amer. Math. Soc. 200 (1974), 369–393. MR 358781, DOI 10.1090/S0002-9947-1974-0358781-0
- Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095, DOI 10.1090/coll/028
Additional Information
- © Copyright 1991 American Mathematical Society
- 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