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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 M with non-cyclic stabilizer Gx 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 2000
Primary: 57S17
Secondary: 57R85
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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