Geometric 4-manifolds in the sense of Thurston and Seifert 4-manifolds I

Abstract

B$ and its underlying space are denoted by $\pi_{1}^{orb}B$ and $|B|$ respectively.We define $[x, y]=xyx^{-1}y^{-1}$ for the elements $x,$ $y\in\pi_{1}S$ .We use the symbols for the closed euclidean 2-orbifolds as indicated in Figure 1.In Figure 1 the cone point of angle $2\pi/m$ and the corner reflector of angle $\pi/m$ are represented by $m$ and $\overline{m}$ respectively.Throughout this paper all the subjects will be considered in the smooth category.\S 1. Seifert 4-manifolds over 2-orbifolds with reflectors.A closed orientable 4-manifold $S$ is called a Seifert 4-manifold if (1) $S$ has a srructure of a fibered orbifold $\pi$ : $Sarrow B$ over a 2-orbifold $B$ with general fiber a 2-torus $T^{2}$ and (2) $S$ is non-singular as an orbifold.$S$ is represented by some invariants analogous to those for Seifert 3-orbifolds with general fiber $S^{1}([3]$ ,[5] $)$ .First we will describe the local pictures of this fibration and then give its global description.LOCAL PICTURES.A point $P$ of $B$ has a neighborhood of type $D=D^{2}/G$ where $D^{2}$ is a 2-disc centered at $O\in R^{2}$ corresponding to $p$ and $G$ is a finite subgroup of 0(2) corresponding to the stabilizer of $p$ .Then $\pi^{-1}(D)$ is identified with $T^{2}\cross D^{2}/G$ where the action of $G$ on $T^{2}xD^{2}$ is free and is some lift of that on $D^{2}$ so that $\pi|\pi^{-1}(D)$ is the map $T^{2}\cross D^{2}/Garrow D^{2}/G$ induced from the natural projection from $T^{2}\cross D^{2}$ to $D^{2}$ .Here $T^{2}$ is identified with $R^{2}/Z^{2}$ and the point of $T^{2}\cross D^{2}$ is represented as $(x, y, z)$ with $(x, y)\in R^{2}(mod Z^{2})$ and $z\in C$ , $z|$ Sl.Let 1 and $h$ be the curves represented by $R/Z\cross\{0\}$ and $\{0\}\cross R/Z$ respectively.Case $0$ .$G=id$ .In this case $p$ is a nonsingular point and the fiber over $p$ is called a general fiber.Case 1. $G=Z_{m}$ where the generator $\rho$ of $Z_{m}$ acts on $T^{2}\cross D^{2}$ by $\rho(x, y, z)=$ $(x-a/m, y-b/m, \exp(2\pi i/m)z)$ with $g$ .$c$ .$d$ .$(m, a, b)=1$ .In this case $P$ is a cone point of angle $2\pi/m$ and the fiber over $P$ is called a multiple torus of type $(m, a, b)$ .Case 2. $G=Z_{2}$ where the generator $f$ of $Z_{2}$ acts on $T^{2}\cross D^{2}$ by $c(x, y, z)=$ $(x+1/2, -y,\overline{z})$ .In this case $P$ is on the reflector and the fiber over $P$ is a Klein bottle $K$ and $\pi^{-1}(D)$ is a twisted $D^{2}$ -bundle over $K$ .Case 3. $G=D_{2m}=\{c, \rho|f^{2}=\rho^{m}=1, c\rho c^{-1}=\rho^{-1}\}$ whose action on $T^{2}\cross D^{2}$ is defined by $\rho(x, y, z)=(x, y-b/m, \exp(2\pi i/m)z),$ $c(x, y, z)=(x+1/2, -y,\overline{z})$ with $g$ .$c$ .$d$ .$(m, b)=1$ .In this case $P$ is a corner reflector of angle $\pi/m$ and the fiber over $p$ is a Klein bottle whose fundamental domain is $1/m$ -times that of the fiber of the reflector point near $p$ .We call this fiber a multiple Klein bottle of type $(m, 0, b)$ .Here we note that the fiber of this tyPe cannot be twisted along 1.

Locations

• Journal of the Mathematical Society of Japan - View - PDF