In the late seventies Segal proved that the homotopy type of the space of continuous map of degree n from the sphere S^2 to itself can be approximated by the space of holomorphic functions of degree n from P^1 to P^1. His work was motivated by the observation of Atiyah that the only critical points of an "energy" functional on the space of continuous maps is the space of rational maps (where the functional achieves an absolute minimum) and the extrapolation of finite dimensional Morse theory to the infinite dimensional case.
Since then there has been a lot of work in the case of a more general target space X. The space of symplectic maps of S^2 to a symplectic manifold X has played a fundamental role in many branches of mathematics, such as analysis, differential geometry, topology and linear control theory. Identifying S^2 with P^1 we prove that the homotopy group of the space of holomorphic maps of degree n is independent of n, when n is large.