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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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