Topology of Hopf surfaces
Masahide Kato · Journal of the Mathematical Society of Japan · 1975
By a Hopf surface $S$ , we shall mean a 2-dimensional compact complex manifold of which the universal covering is $W=C^{2}-\{0\}$ , where $C^{2}$ is the space of two complex variables $(z_{1}, z_{2})$ and $0$ is the origin $(0,0)$ .$S$ can be represented as a quotient space $W/G$ with a group $G$ generated by some biholomorphic transformations of $W$ whose action is properly discontinuous and free.Let $B$ be a closed spherical set in $C^{2}$ determined by the inequality:By Kodaira [4], $G$ has the following properties;(1) $G$ contains a contraction $g$ , and the infinite cyclic subgroup generated by $g$ has a finite index in $G$ , (2) if $G$ is non-abelian, then by a Proper choice of global coordinates of $C^{2},$ $G$ appears as a subgroup of $GL(2, C)$ .The purpose of this paper is to classify all Hopf surfaces by diffeomor- phisms (or equivalently, homeomorphisms).(See, Theorems 9, 10 and 12.) \S 2. Classification of $G$ in the case $G\subset GL(2, C)$ .First we define subgroups $H$ and $K$ of $G$ as follows;Clearly, $G\triangleright H\triangleright K$ and $G\triangleright K$ .In what follows $H$ and $K$ are assumed to satisfy these conditions.Let $G_{2}$ be a subgroup of a group $G_{1}$ .We denote by $[G_{1} : G_{2}]$ the index of $G_{2}$ in $G_{1}$ .If $G_{2}$ is generated by some elements $h_{1},$ $\cdots$ , $h_{r}$ of $G_{1}$ , we sometimes write $\{h_{1}, \cdots , h_{r}\}$ instead of $G_{2}$ .LEMMA 1.An element $x$ of $G$ is a contraction if and only if $|\det x||\beta|$ and $|\beta|\geqq 1>|\alpha|$ .Hence we may assume that $|\alpha|\geqq 1>|\beta|$ .Then $x$ is of the infinite order and not a contraction.Let $g$ be a contraction in $G$ .Then we have $\{x\}\cap\{g\}=\{1\}$ .This implies that $[G:\{g\}]=\infty$ .