GEOMETRIC APPROXIMATION OF THE FIBRE OF THE FREUDENTHAL SUSPENSION

Abstract

Let Rat_k(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:

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

The study of the topology of Rat_k(CPⁿ) originated in [10]. Later, the stable homotopy type of Rat_k(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 Rat_k(CPⁿ)→ Ω²_kCPⁿ≃ Ω²S²ⁿ⁺¹ is a homotopy equivalence up to dimension k(2n−1). Thus if we form the direct limit Rat_∞(CPⁿ)= lim_k→∞ Rat_k(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 X_k(CPⁿ), which are natural generalizations of Rat_k(CPⁿ), so that X_∞(CPⁿ) approximates W(S²ⁿ). Moreover, we study the stable homotopy type of X_k(CPⁿ).

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