Actions on lens spaces which respect a Heegaard decomposition
For \(L= L(p,q)\) a 3-dimensional lens space, the authors consider smooth finite group actions on \(L\) which leave a Heegaard torus and each of its complementary components invariants. In this paper they enumerate and classify two type of actions (rotational and dirotational) up to equivalence (two actions are equivalent if their images are conjugate in the group of self-diffeomorphisms of \(L\)). First, they establish the geometrization and rigidity for actions which respect a Heegaard decomposition. Then the results are combined to describe the classification of the equivalence classes of actions of rotational type. They also examine the fixed point data of each equivalence of cyclic actions of rotational type. Finally, in the main result of this paper, going beyond the geometrization and rigidity conjectures they give a complete enumeration of the equivalence classes of group actions on \(L\) which respect a Heegaard decomposition. Using recent progress on Thurston's geometrization conjecture these results give more complete solutions to the classification problem of symmetries on lens spaces.
scientific article; zbMATH DE number 1917841