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.
1980 Mathematics subject classifications
Primary 55P15 58D15 Keywords Homotopy type component space of maps between surfaces self maps on the 2-sphere
This is a preview of subscription content, log in to check access.