Algebraic & Geometric Topology 3 (2003) 709–718

DOI: 10.2140/agt.2003.3.709

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 Szucs 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(3).

Keywords

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

Mathematical Subject Classification

Primary: 57S17

Secondary: 57R85

References

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Publication

Received: 7 January 2003
Accepted: 14 July 2003
Published: 30 July 2003

Authors