#### 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.

Notes on file formats
Peter E. Frenkel

Department of Geometry, Mathematics Institute

Budapest University of Technology and Economics, Egry J. u. 1.

1111 Budapest, Hungary

Email: frenkelp@renyi.hu

AGT home page

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