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.

Finite, Fiber- and Orientation-Preserving Group Actions on Totally Orientable Seifert Manifolds

“…Note that the quotient orbifold B 1 /ϕ 1 necessarily has boundary either S 2 (n, n) or S 2 (2, 2, n). This follows from John Kalliongis and Ryo Ohashi in their paper Finite actions on the 2-sphere, the projective plane and I-bundles over the projective plane [11], where they show that these are the only orientable quotients of S 2 where the action fixes one point or exchanges two points (corresponding to the two discs D 1 , D 2 ).…”

“…In the particular case of the base space being S 2 this is not a restriction as any finite group that acts is a subgroup of a finite group that has this property. For clarification of this see again [11].…”

“…Any action on F can only exchange two of the boundary components and permute the remaining three. Referring to John Kalliongis and Ryo Ohashi's paper Finite actions on the 2-sphere, the projective plane and I-bundles over the projective plane [11], we learn that we need the group to be a subgroup of a group of the form Dih(Z 3 ) generated by an order three rotation that fixes two boundary components and either an order two rotation or a relection.…”