Geodesic
Fixed point data of finite groups acting on 3–manifolds
Frenkel, Peter E1
1Department of Geometry, Mathematics Institute, Budapest University of Technology and Economics, Egry J. u. 1., 1111 Budapest, Hungary
Algebraic and Geometric Topology, Tome 3 (2003) no. 2, pp. 709-718
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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).

DOI : 10.2140/agt.2003.3.709
Keywords: 3–manifold, group action, fixed points, equivariant cobordism

Frenkel, Peter E  1

1 Department of Geometry, Mathematics Institute, Budapest University of Technology and Economics, Egry J. u. 1., 1111 Budapest, Hungary 
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Frenkel, Peter E. Fixed point data of finite groups acting on 3–manifolds. Algebraic and Geometric Topology, Tome 3 (2003) no. 2, pp. 709-718. doi: 10.2140/agt.2003.3.709

[1] T Tom Dieck, Characteristic numbers of $G$–manifolds I, Invent. Math. 13 (1971) 213

[2] S S Khare, Finite group action and equivariant bordism, Pacific J. Math. 116 (1985) 39

[3] C Kosniowski, Some equivariant bordism theories vanish, Math. Ann. 242 (1979) 59

[4] C Kosniowski, Fixed points and group actions, from: "Algebraic topology, Aarhus 1982 (Aarhus, 1982)", Lecture Notes in Math. 1051, Springer (1984) 603

[5] R E Stong, Equivalent bordism and $(\mathbb{Z}_{2})^{k}$ actions, Duke Math. J. 37 (1970) 779

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