Rational Functions, Labelled Configurations, And Hilbert Schemes

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Journal Of The London Mathematical Society


In this paper, we continue the study of the homotopy type of spaces of rational functions from S2 to CP(n) begun in [3, 4]. We prove that, for n > 1, Rat(k)(CP(n)) is homotopy equivalent to C(k)(R2,S2n-1), the configuration space of distinct points in R2 with labels in S2n-1 of length at most k. This desuspends the stable homotopy theoretic theorems of [3, 4]. We also give direct homotopy equivalences between C(k)(R2,S2n-1) and the Hilbert scheme moduli space for Rat(k)(CP(n)) defined by Atiyah and Hitchin [1]. When n = 1, these results no longer hold in general, and, as an illustration, we determine the homotopy types of Rat2(CP1) and C2(R2,S1) and show how they differ.