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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.

##### MSC:
 55P15 Classification of homotopy type 55Q52 Homotopy groups of special spaces 55P10 Homotopy equivalences in algebraic topology 55P35 Loop spaces
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