The homotopy type of rational functions
Let \(\text{Rat}_ k(\mathbb{C}\mathbb{P}^ n)\) denote the space of degree \(k\), basepoint preserving, holomorphic (equivalently, rational) maps from the Riemann sphere \(S^ 2\) to complex projective space \(\mathbb{C}\mathbb{P}^ n\). In this note we give a relatively short, combinatorial proof that \(\text{Rat}_ k (\mathbb{C}\mathbb{P}^ n)\) is stably homotopy equivalent to a wedge of subquotients of the May-Milgram filtered model for the double loop space of the 3-sphere, \(\Omega^ 2 S^ 3\). This implies that, when \(n=1\), \(\text{Rat}_ k(\mathbb{C}\mathbb{P}^ 1)\) is stably homotopy equivalent to \(K(\beta_{2k},1)\), the Eilenberg-MacLane space associated to Artin's braid group on \(2k\) strings. A more complicated homological proof of this theorem that extends to other mapping spaces and various applications are discussed in our companion paper [Acta Math. 166, No. 3/4, 163-221 (1991; Zbl 0741.55005)].
scientific article; zbMATH DE number 549778