Fixed point data of finite groups acting on 3–manifolds

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 $S O (3)$ .

Keywords

3–manifold, group action, fixed points, equivariant cobordism

Mathematical Subject Classification 2000

Primary: 57S17

Secondary: 57R85

References

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Publication

Received: 7 January 2003

Accepted: 14 July 2003

Published: 30 July 2003

Authors

Peter E Frenkel
Department of Geometry Mathematics Institute Budapest University of Technology and Economics Egry J. u. 1. 1111 Budapest Hungary

Volume 3, issue 2 (2003)

Peter E Frenkel