Let Ratk(CPn) denote the space of based holomorphic maps of degree k from the Riemannian sphere S2 to the complex projective space CPn. The basepoint condition we assume is that f(∞)=[1, …, 1]. Such holomorphic maps are given by rational functions:

Ratk(CPn) ={(p0(z), …, pn(z))[ratio ]each pi(z) is a monic, degree-k polynomial and such that there are no roots common to all pi(z)}. (1.1)

The study of the topology of Ratk(CPn) originated in [10]. Later, the stable homotopy type of Ratk(CPn) was described in [3] in terms of configuration spaces and Artin's braid groups. Let W(S2n) denote the homotopy theoretic fibre of the Freudenthal suspension E[ratio ]S2n→ ΩS2n+1. Then we have the following sequence of fibrations: Ω2S2n+1→ W(S2n)→S2n→ ΩS2n+1. A theorem in [10] tells us that the inclusion Ratk(CPn)→ Ω2kCPn≃ Ω2S2n+1 is a homotopy equivalence up to dimension k(2n−1). Thus if we form the direct limit Rat∞(CPn)= limk→∞ Ratk(CPn), we have, in particular, that Rat∞(CPn) is homotopy equivalent to Ω2S2n+1.

If we take the results of [3] and [10] into account, we naturally encounter the following problem: how to construct spaces Xk(CPn), which are natural generalizations of Ratk(CPn), so that X∞(CPn) approximates W(S2n). Moreover, we study the stable homotopy type of Xk(CPn).

The purpose of this paper is to give an answer to this problem. The results are stated after the following definition.