Geometric Approximation Of The Fibre Of The Freudenthal Suspension

Let Rat k ( C P n ) denote the space of based holomorphic maps of degree k from the Riemannian sphere S 2 to the complex projective space C P n . The basepoint condition we assume is that f (∞)=[1, …, 1]. Such holomorphic maps are given by rational functions: Rat k ( C P n ) ={( p 0 ( z ), …, p n ( z )):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 ( C P n ) originated in [ 10 ]. Later, the stable homotopy type of Rat k ( C P n ) was described in [ 3 ] in terms of configuration spaces and Artin's braid groups. Let W ( S 2 n ) denote the homotopy theoretic fibre of the Freudenthal suspension E : S 2 n → Ω S 2 n +1 . Then we have the following sequence of fibrations: Ω 2 S 2 n +1 → W ( S 2 n )→ S 2 n → Ω S 2 n +1 . A theorem in [ 10 ] tells us that the inclusion Rat k ( C P n )→ Ω 2 k C P n ≃ Ω 2 S 2 n +1 is a homotopy equivalence up to dimension k (2 n −1). Thus if we form the direct limit Rat ∞ ( C P n )= lim k →∞ Rat k ( C P n ), we have, in particular, that Rat ∞ ( C P n ) is homotopy equivalent to Ω 2 S 2 n +1 . If we take the results of [ 3 ] and [ 10 ] into account, we naturally encounter the following problem: how to construct spaces X k ( C P n ), which are natural generalizations of Rat k ( C P n ), so that X ∞ ( C P n ) approximates W ( S 2 n ). Moreover, we study the stable homotopy type of X k ( C P n ). The purpose of this paper is to give an answer to this problem. The results are stated after the following definition. 1991 Mathematics Subject Classification 55P35 .