Finite actions on the 2-sphere, the projective plane and I-bundles over the projective plane

Abstract

In this paper, we consider the finite groups which act on the 2 -sphere S 2 and the projective plane P 2 , and show how to visualize these actions which are explicitly defined. We obtain their quotient types by distinguishing a fundamental domain for each action and identifying its boundary. If G is an action on P 2 , then G is isomorphic to one of the following groups: S 4 , A 5 , A 4 , Z m or Dih( Z m ) . For each group, there is only one equivalence class (conjugation), and G leaves an orientation reversing loop invariant if and only if G is isomorphic to either Z m or Dih( Z m ) . Using these preliminary results, we classify and enumerate the finite groups, up to equivalence, which act on P 2 × I and the twisted I-bundle over P 2 . As an example, if m > 2 is an even integer and m /2 is odd, there are three equivalence classes of orientation reversing Dih( Z m ) -actions on the twisted I-bundle over P 2 . However if m /2 is even, then there are two equivalence classes.