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 S² and the projective plane P², 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 P², then G is isomorphic to one of the following groups: S₄, A₅, A₄, 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 P² × I and the twisted I-bundle over P². 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 P². However if m/2 is even, then there are two equivalence classes.