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

John Kalliongis, Ryo Ohashi

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, 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.


Keywords


Achiral symmetry, chiral symmetry, equivalence of actions, finite group action, isometry, orbifold, symmetry

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DOI: https://doi.org/10.26493/1855-3974.806.c9d

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications