Group actions on spaces of rational functions
This paper continues and extends previous work of the second-named author and others on the homotopy type of certain kinds of function spaces. Let \(\text{Hol}_d\) denote the space of all holomorphic maps of degree \(d\geq 0\) from the Riemann sphere \(S^2\) to itself. Then for each \(d\) there is a corresponding evaluation fibration sequence \(\text{Hol}^*_d \to \text{Hol}_d \to S^2\) with fibre \(\text{Hol}^*_d\) consisting of basepoint-preserving holomorphic maps. The function spaces \(\text{Hol}_d\) and \(\text{Hol}^*_d\) are of interest from a number of points of view and many results are known concerning the homotopy type of these function spaces. One connection with other areas of mathematics arises as follows: Denote the orbit space \(\text{Hol}_1\backslash\text{Hol}_d\) of the obvious action of \(\text{Hol}_1\) on \(\text{Hol}_d\) by \(X_d\). Then a theorem of Milgram says that, for \(d \geq 1\), \(X_d\) is homeomorphic to the space of non-singular \(d \times d\) Toeplitz matrices. Here, the authors focus on the homotopy type of the universal covers \(\widetilde{\text{Hol}}_d\) and \(\widetilde{\text{Hol}^*_d}\). Their main results give homotopy equivalences as follows for \(d\geq 1\): \(\widetilde{\text{Hol}}_d \simeq S^3 \times \widetilde{X}_d\) (Theorem 1.4), and \(\widetilde{X}_d \simeq \widetilde{\text{Hol}^*_d}\) (Theorem 1.5). From these results, the homotopy equivalence \(\widetilde{\text{Hol}}_d \simeq S^3 \times \widetilde{\text{Hol}^*_d}\) is evident and the isomorphisms \(\pi_k(\text{Hol}_d) \cong \pi_k(S^3) \oplus \pi_{k+2}(S^2)\), for \(2 \leq k < d\) may be obtained. The last two consequences are also obtained in [\textit{K.Yamaguchi}, Kyushu J. Math. 56, 381-387 (2002; Zbl 1041.55005)]. In the case \(d=2\), the homotopy types of \(\text{Hol}_2\), \(\widetilde{\text{Hol}_2}\), and \(\widetilde{\text{Hol}^*_2}\) have been explicitly identified as homogeneous spaces in [\textit{M. Guest, A. Kozlowski, M. Murayama}, and \textit{K. Yamaguchi}, J. Math. Kyoto Univ. 35, 631-638 (1995; Zbl 0862.55011)]. This latter paper also contains computations of some homotopy groups \(\pi_k(\text{Hol}_d)\), which are obtained again in the paper under review. In these previous papers, the main tool -- at least for the homotopy calculations -- was the evaluation fibration sequence \(\text{Hol}^*_d \to \text{Hol}_d \to S^2\) and its interplay with the evaluation fibration sequence \(\text{Map}^*_d \to \text{Map}_d \to S^2\) obtained by considering continuous maps of degree \(d\) from \(S^2\) to itself. In the paper under review, on the other hand, the results flow from a study of the action of \(\text{Hol}_1\) on \(\text{Hol}_d\). That this approach is fruitful is due in large part to the authors' skill in drawing on a wealth of facts and previous results in this area.
scientific article; zbMATH DE number 1917053