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, Zm or Dih(Zm). 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 Zm or Dih(Zm). 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(Zm)-actions on the twisted I-bundle over P2. However if m/2 is even, then there are two equivalence classes.
@article{806,
title = {Finite actions on the 2-sphere, the projective plane and I-bundles over the projective plane},
journal = {ARS MATHEMATICA CONTEMPORANEA},
volume = {15},
year = {2018},
doi = {10.26493/1855-3974.806.c9d},
language = {EN},
url = {http://dml.mathdoc.fr/item/806}
}
Kalliongis, John; Ohashi, Ryo. Finite actions on the 2-sphere, the projective plane and I-bundles over the projective plane. ARS MATHEMATICA CONTEMPORANEA, Tome 15 (2018) . doi : 10.26493/1855-3974.806.c9d. http://gdmltest.u-ga.fr/item/806/