zbMATH — the first resource for mathematics

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 \(\bigsqcup_{d \geq 0} \text{Hol}_d\).

55Q52 Homotopy groups of special spaces
57R35 Differentiable mappings in differential topology
Full Text: DOI