Abstract
In this paper we review contributions to the homotopy theory of manifolds of maps between closed orientable surfaces, and in particular those results which provide a full homotopy type of a component. As a main case, we describe the complete homotopy type of the space of orientation preserving self homotopy equivalences on the 2-sphere (the component containing the maps of degree 1) in terms of well known spaces in topology. As a new result, we prove that the component in the space of self maps on the 2-sphere containing the maps of degree k admits a unique k-fold covering space, and that this covering space has the homotopy type of the space of orientation preserving self homotopy equivalences.
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Bibliography
R. Geoghegan, On spaces of homeomorphisms, embeddings and functions-I, Topology 11 (1972), 159–177.
D.H. Gottlieb, A certain subgroup of the fundamental group, Amer.J. Math. 87 (1965), 840–856.
D.H. Gottlieb, Covering transformations and universal fibrations, Illinois J. Math. 13 (1969), 432–437.
V.L. Hansen, On the space of maps of a closed surface into the 2-sphere, Math. Scand. 35 (1974), 149–158.
V.L. Hansen, On a theorem of Al'ber on spaces of maps, J. Diff. Geom. 12 (1977), 565–566.
V.L. Hansen, Decomposability of evaluation fibrations and the brace product operation of James, Compositio Math. 35 (1977), 83–89.
V.L. Hansen, Spaces of maps into Eilenberg-MacLane spaces, Canadian J. Math. XXXIII (1981), 782–785.
V.L. Hansen, The homotopy groups of a space of maps between oriented closed surfaces, Bull. London Math. Soc. 15 (1983), 360–364.
V.L. Hansen, On Steenrod bundles and the van Kampen theorem, Canadian Math. Bull. 31 (1988), 241–249.
D.W. Henderson, Stable classification of infinite dimensional manifolds by homotopy type, Invent. Math. 12 (1971), 48–56.
S.T. Hu, Concerning the homotopy groups of the components of the mapping space Y Sp, Indagationes Math. 8 (1946), 623–629.
H. Kneser, Die Deformationssätze der einfach zusammenhängenden Flächen, Math. Z. 25 (1926), 362–372.
L.L. Larmore and E. Thomas, On the fundamental group of a space of sections, Math. Scand. 47 (1980), 232–246.
S. Smale, Diffeomorphisms on the 2-sphere, Proc. Amer. Math. Soc. 10 (1959), 621–626.
G.W. Whitehead, On products in homotopy groups, Ann. Math. 47 (1946), 460–475.
J.H.C. Whitehead, On certain theorems of G.W. Whitehead, Ann. Math. 58 (1953), 418–428.
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© 1990 Springer-Verlag
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Hansen, V.L. (1990). The space of self maps on the 2-sphere. In: Piccinini, R.A. (eds) Groups of Self-Equivalences and Related Topics. Lecture Notes in Mathematics, vol 1425. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083829
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DOI: https://doi.org/10.1007/BFb0083829
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