On the fundamental group of the space of harmonic 2-spheres in then-sphere

Abstract

Harmonic maps φ : S −→ S are of interest from the point of view of differential geometry (as branched minimal immersions), and from the point of view of gauge theory (as classical solutions of the S non-linear sigma model). From the work of Chern, Calabi, Barbosa and others, there is now a complete description of such harmonic maps in terms of holomorphic data. However, very little is known about the global properties of the space Harm(S, S) of all harmonic maps. Since the analogous object in Yang-Mills theory — the space of solutions to the Yang-Mills equations — has proved to be of great significance, the space of harmonic maps is also of some interest. In this paper we shall give an approach to the study of the topology of Harm(S, S).