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Group actions on spaces of rational functions. (English) Zbl 1026.55011
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 [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 [M. Guest, A. Kozlowski, M. Murayama, and 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.

55P15 Classification of homotopy type
55Q52 Homotopy groups of special spaces
55P10 Homotopy equivalences in algebraic topology
55P35 Loop spaces
Full Text: DOI
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