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)

 
 

 

The space of retractions of the $ 2$-sphere and the annulus


Author: Neal R. Wagner
Journal: Trans. Amer. Math. Soc. 158 (1971), 319-329
MSC: Primary 54.28
DOI: https://doi.org/10.1090/S0002-9947-1971-0279763-0
MathSciNet review: 0279763
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a manifold $ M$, there is an embedding $ \Lambda $ of $ M$ into the space of retractions of $ M$, taking each point to the retraction of $ M$ to that point. Considering $ \Lambda $ as a map into the connected component containing its image, we prove that $ \Lambda $ is a weak homotopy equivalence for two choices of $ M$, namely, the $ 2$-sphere and the annulus.


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

  • [1] J. W. Alexander, On the deformation of an $ n$-cell, Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 406-407.
  • [2] K. Borsuk, Theory of retracts, Monografie Mat., Tom 44, PWN, Warsaw, 1967. MR 35 #7306. MR 0216473 (35:7306)
  • [3] -, Concerning the set of retractions, Colloq. Math. 18 (1967), 197-201. MR 36 #2126. MR 0219043 (36:2126)
  • [4] R. Courant, Über eine Eigenschaft der Abbildungsfunctionen bei konformer Abbildung, Gött. Nachr. (1914), 101-109.
  • [5] -, Bemerkung zu meiner Note: ``Über eine Eigenschaft...", Gött. Nachr. (1922), 69-70.
  • [6] -, Dirichlet's principle, conformal mapping, and minimal surfaces, Interscience, New York, 1950. MR 12, 90.
  • [7] E. Dyer and M.-E. Hamstrom, Completely regular mappings, Fund. Math. 45 (1958), 103-118. MR 19, 1187. MR 0092959 (19:1187e)
  • [8] D. B. A. Epstein, Curves on $ 2$-manifolds and isotopies, Acta Math. 115 (1966), 83-107. MR 35 #4938. MR 0214087 (35:4938)
  • [9] C. Gattegno and A. Ostrowski, Représentation conforme a la frontière; domaines généraux, Mém. Sci. Math., no. 109, Gauthier-Villars, Paris, 1949. MR 11, 425.
  • [10] M.-E. Hamstrom, Some global properties of the space of homeomorphisms on a disc with holes, Duke Math. J. 29 (1962), 657-662. MR 26 #745. MR 0143185 (26:745)
  • [11] M.-E. Hamstrom and E. Dyer, Regular mappings and the space of homeomorphisms on a $ 2$-manifold, Duke Math. J. 25 (1958), 521-531. MR 20 #2695. MR 0096202 (20:2695)
  • [12] S.-T. Hu, Homotopy theory, Pure and Appl. Math., vol. 8, Academic Press, New York, 1959. MR 21 #5186. MR 0106454 (21:5186)
  • [13] A. I. Markuševič, Theory of functions of a complex variable. Vol. III, GITTL, Moscow, 1950; English transl., Prentice-Hall, Englewood Cliff's, N. J., 1967. MR 12, 87; MR 35 #6799. MR 0215964 (35:6799)
  • [14] E. Michael, Continuous selections. II, Ann. of Math. (2) 64 (1956), 562-580. MR 18, 325. MR 0080909 (18:325e)
  • [15] H. R. Morton, The space of homeomorphisms of a disc with $ n$ holes, Illinois J. Math. 11 (1967), 40-48. MR 34 #5066. MR 0205233 (34:5066)
  • [16] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54.28

Retrieve articles in all journals with MSC: 54.28


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0279763-0
Keywords: Retract, retraction, two-manifold, two-sphere, annulus, homotopy equivalence, weak homotopy equivalence, function space, compact-open topology, selection
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society