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Rational functions, labelled configurations, and Hilbert schemes. (English) Zbl 0756.55005
The authors study the homotopy type of spaces of rational functions from $$S^ 2$$ to $$\mathbb{C} P^ n$$. The authors prove that, for $$n>1$$, $$\text{Rat}_ k(\mathbb{C} P^ n)$$ is homotopy equivalent to $$C_ k(\mathbb{R}^ 2,S^{2n-1})$$, the configuration space of distinct points in $$\mathbb{R}^ 2$$ with labels in $$S^{2n-1}$$ of length at most $$k$$. The authors also give direct homotopy equivalences between $$C_ k(\mathbb{R}^ 2,S^{2n-1})$$ and the Hilbert scheme moduli space for $$\text{Rat}_ k(\mathbb{C} P^ n)$$ defined by M. Atiyah and N. Hitchin [The geometry and dynamics of magnetic monopoles (1988; Zbl 0671.53001)]. When $$n=1$$, these results no longer hold in general, and, as an illustration, the authors give the homotopy types of $$\text{Rat}_ 2(\mathbb{C} P^ 1)$$ and $$C_ 2(\mathbb{R}^ 2,S^ 1)$$ and show how they differ.
The autors announce two other papers on the same subject [F. R. Cohen, R. L. Cohen, B. M. Mann and R. J. Milgram, “The topology of rational functions and divisors of surfaces”, Acta Math. 166, No. 3/4, 163-221 (1991), “The homotopy type of rational functions”, Math. Z. (to appear)].
Reviewer: I.Pop (Iaşi)

##### MSC:
 55P35 Loop spaces
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