Fixed point data of finite groups acting on 3–manifolds

Abstract

We consider fully effective orientation-preserving smooth actions of a given finite group $G$ on smooth, closed, oriented 3–manifolds $M$. We investigate the relations that necessarily hold between the numbers of fixed points of various non-cyclic subgroups. In Section 2, we show that all such relations are in fact equations mod 2, and we explain how the number of independent equations yields information concerning low-dimensional equivariant cobordism groups. Moreover, we restate a theorem of A Szűcs asserting that under the conditions imposed on a smooth action of $G$ on $M$ as above, the number of $G$–orbits of points $x\in M$ with non-cyclic stabilizer $G_x$ is even, and we prove the result by using arguments of G Moussong. In Sections 3 and 4, we determine all the equations for non-cyclic subgroups $G$ of $SO\left(3\right)$.