#### Volume 3, issue 2 (2003)

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 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Fixed point data of finite groups acting on 3–manifolds

### Peter E Frenkel

Algebraic & Geometric Topology 3 (2003) 709–718
 arXiv: math.AT/0301159
##### 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 $SO\left(3\right)$.

##### Keywords
3–manifold, group action, fixed points, equivariant cobordism
Primary: 57S17
Secondary: 57R85