Fixed point data of finite groups acting on 3-manifolds

Details Fixed point data of finite groups acting on 3-manifolds Fixed point data of finite groups acting on 3-manifolds

2003-01-15

arxiv.org/abs/math/0301159v2

Algebr. Geom. Topol. 3 (2003) 709-718

Mathematics

Algebraic Topology

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-24.abs.html

Scientific paper

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).

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