Kohhei Yamaguchi

Connective coverings of spaces of holomorphic maps

For each integer ≥ 0, we denote by Hol (2 CP ) the space consisting of all holomorphic maps 2 → CP of degree . The corresponding space of continuous maps is denoted by Map ( 2 CP ). We also denote by Hol ( 2 CP ) (resp. 2CP ) the subspace of Hol (2 CP ) (resp. Map (2 CP )) consisting of all maps ∈ Hol ( 2 CP ) which preserve the base-points. The space of holomorphic maps are of interest both from a classical and modern point of view (e.g. [1], [3], [6]). It is an elementary and fundamental fact that Hol ( 2 CP ) and Hol ( 2 CP ) are connected spaces. If = 1, the fundamental groups of these spaces are Z/2 and Z, repectively ([7], [12]); if ≥ 2, these spaces are simply connected and 2( − 1)-connected, respectively. The following more general result was obtain ed by G. Segal: