Algebraic and Geometric Topology 3 (2003), paper no. 24, pages 709-718.

Fixed point data of finite groups acting on 3-manifolds

Peter E. Frenkel

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

AMS subject classification. Primary: 57S17. Secondary: 57R85.

DOI: 10.2140/agt.2003.3.709

E-print: arXiv:math.AT/0301159

Submitted: 7 January 2003. Accepted: 14 July 2003. Published: 30 July 2003.

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

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