Fundamental groups of spaces of holomorphic maps and group actions
Let \(Hol_d(\mathbb CP^k,\mathbb CP^n)\) denote the space of holomorphic maps \(\mathbb CP^k\to \mathbb CP^n\) of degree \(d\), \(d\geq 0\), \(1\leq k\leq n\). Let \(map_d(\mathbb CP^k,\mathbb CP^n)\) be the analogous space of continuous maps, where the degree is defined as the effect on \(\pi_2\). There is an obvious inclusion as well as similar statements for pointed maps. \textit{G. B. Segal} [Acta Math. 143, 39--72 (1979; Zbl 0427.55006)] showed that the inclusions are homotopy equivalences up to certain dimensions, and \textit{R. J. Milgram} [Topology 36, 1155--1192 (1997; Zbl 0894.57016)] and \textit{J. M. Møller} [Proc. Am. Math. Soc. 91, 471--476 (1984; Zbl 0514.55011)] contributed to our understanding of these spaces and their quotients under the action of projective linear groups. The present paper focuses on fundamental groups. If \(k<n\), \(Hol_d(\mathbb CP^k,\mathbb CP^n)\) and its pointed equivalent are simply-connected. But \(\pi_1(Hol_d(\mathbb CP^n,\mathbb CP^n)) = \mathbb Z/(n+1)d^n\). There are many other interesting results on fundamental groups and on the fibre space techniques that are used to calculate properties of such spaces.
scientific article; zbMATH DE number 5010259