From: rong@math.gwu.edu (Yongwu Rong) Newsgroups: sci.math.research,sci.math Subject: Re: A 3-manifold question Date: 22 Aug 1994 14:15:06 GMT In article <32obpe\$i2j@cronkite.seas.gwu.edu>, > >This question is more or less equivalent to: determine all >periodic maps up to conjugacy on a given surface. > I tried to post the following last week but it didn't work, so I am giving another shot. Y.R. 8/22 %%%%%%%%%% text written on 8/17 After some email contact with Charlie Frohman and Dan Asimov, plus some patient work last night, I now believe I have a solution to my question. Below is a complete (I hope) list of periodic homeomorphisms of the genus 2 surface. There are nine such maps up to some equivalence relations such as conjugacy, or taking a power that is relatively prime to the order. Here F denotes the genus 2 surface, f: F -> F is the periodic homeomorphism of F with order d, and G is the quotient orbifold. 1. d=2, G = T(2,2), f is a \pi rotation of F along the "short axis". f has two fixed points. 2. d=2, G = S(2,2,2,2,2,2), f is a \pi rotation along the "long axis". f has 6 fixed points, and f is often called the mutation map. 3. d=3, G = S(3,3,3,3), to see f, consider two concentric spheres connected by 3 tubes that are 120 degree apart. The resulting surface is F. Now rotate F 120 degree gives the map f with order 3. f has 4 fixed points. 4. d=4, G = S(2,2,4,4), to see f, consider the standard 8-gon representation of F by identify pairs of edges. A 90 degree rotation of the 8-gon induces the order 4 map f. f has 2 fixed points, and 2 periodic orbits each of length 2. 5. d=5, G = S(5,5,5), cut the above 8-gon into two pentagons, and rotate each by a 72 degree rotation, inducing f. f has 3 fixed points. 6. d=6, G = S(3,6,6), I don't know what is the best way to describe f, but G is a 2-fold quotient of the S(3,3,3,3) (d=3 case). To see this put the 4 cone points in S(3,3,3,3) symmetrically in East, South, West, North. Now rotation S(3,3,3,3) 180 degree along the East-West axis gives S(3,6,6). The order 6 map f on F should be the lift of the involution on F. f has 2 fixed points, and 1 periodic orbits of length 2. 7. d=6, G = S(2,2,3,3), f is the product of the f in S(3,3,3,3) (d=3 case) and the mutation (d=2, S(2,2,2,2,2,2) case). The mutation can be seen in the "two-concentric-spheres-plus-3-tubes" representation of F as follows: simply interchange the inner sphere with the outer one, and flip the 3 tubes. f has no fixed points, but has a few periodic orbits. 8. d=8, G = S(2,8,8), f is induced by a 45 degree rotation of the 8-gon, (identified by the word: a b c d a^-1 b^-1 c^-1 d^-1) f has 2 fixed points, and 1 periodic orbits of length 4. 9. d=10, G = S(2,5,10), f is the product of the order 5 map and the mutation. f has 1 fixed point, and 1 periodic orbit of length 2, and one with length 5. There are a lot of relations among these maps. For example the square of an order 6 map is the order 3 map. The main idea in making the list is: first using the Rieman-Hurwitz formula to eliminate possibilities of G, then remove the cone points and get genuine covering from F - disks to G - disks. Covering space theory says this is determined by representations from pi_1 of G - disks to the symmetric group S_d. Also note that covering transformation group is a cyclic group generated by f. All these eliminates more possibilities of G and gives the description of the covering transformation. - Yongwu Rong, George Washington University