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The homotopy type of rational functions. (English) Zbl 0790.55005
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)].

MSC:
 55P42 Stable homotopy theory, spectra 55P45 $$H$$-spaces and duals
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References:
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