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Universal coverings of spaces of holomorphic maps. (English) Zbl 1041.55005
Let $$\text{Hol}_d$$ denote the space of all holomorphic self-maps of the Riemann sphere of degree $$d$$. Let $$\text{Hol}^*_d$$ denote the subspace consisting of basepoint-preserving maps. Let $$\widetilde{\text{Hol}}_d$$ and $$\widetilde{\text{Hol}}^*_d$$ denote their universal covers. The main result is that there is a homotopy equivalence $$\widetilde{\text{Hol}}_d\simeq \widetilde{\text{Hol}}^*_d\times S^3$$. Combined with known results $$\pi_1(\text{Hol}_d)= \mathbb{Z}/2d$$, $$\pi_1(\text{Hol}^*_d)= \mathbb{Z}$$, and $$\text{Hol}^*_d\to \Omega^2_d S^2$$ a homotopy equivalence up to dimension $$d$$, this yields information about $$\text{Hol}_d$$.

##### MSC:
 55P15 Classification of homotopy type 58D15 Manifolds of mappings 32C18 Topology of analytic spaces
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