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The homotopy type of the space of rational functions. (English) Zbl 0862.55011
The authors study the space of all holomorphic self maps of degree \(d \) on the Riemannian 2-sphere \(S^2= \mathbb{C} \cup\infty\). This space is denoted \(\text{Hol}_d \), and \(\text{Hol}^*_d\) denotes those maps preserving a base point. Graeme Segal showed that \(\pi_k (\text{Hol}_d) \cong \pi_k (\text{Map}_d)\) if \(k<d\) where \(\text{Map}_d\) is the space of self maps of \(S^2\). Similarly, \(\pi_k (\text{Hol}^*_d) \cong \pi_k (\text{Map}^*_k) \cong \pi_{k+2} (S^2)\). The authors calculate the homotopy groups for \(\text{Hol}_d\) in terms of the homotopy groups for \(S^3\) and \(S^2\) for \(k \geq 2\) and \(d=1\) and \(d=2\). For \(d \geq 3\) and \(k=2\), the authors prove \(\pi_k (\text{Hol}_d) \cong \pi/2\). Also, if \(d>k \geq 3\), then \(\pi_k (\text{Hol}_d) \cong \pi_{k+2} (S^2) \oplus \pi_k(S^3)\). The authors identify \(\text{Hol}_2 \) and \(\text{Hol}^*_2\) with certain homogeneous spaces. Also they study, as an application, the operated structure on \(\bigsqcup_{d \geq 0} \text{Hol}_d\).

MSC:
55Q52 Homotopy groups of special spaces
57R35 Differentiable mappings in differential topology
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