zbMATH — the first resource for mathematics

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)

55P35 Loop spaces
Full Text: DOI