G-Actions on Riemann Surfaces and the associated Group of Singular Orbit Data

Details G-Actions on Riemann Surfaces and the associated Group of Singular Orbit Data G-Actions on Riemann Surfaces and the associated Group of Singular Orbit Data

1999-02-06

arxiv.org/abs/math/9902048v1

Mathematics

Algebraic Topology

23 pages. See also http://www.math.nwu.edu/~ralph/

Scientific paper

Let $G$ be a finite group. To every smooth $G$-action on a compact, connected and oriented Riemann surface we can associate its data of singular orbits. The set of such data becomes an Abelian group $B_G$ under the $G$-equivariant connected sum. The map which sends $G$ to $B_G$ is functorial and carries many features of the representation theory of finite groups. In this paper we will give a complete computation of the group $B_G$ for any finite group $G$. There is a surjection from the $G$-equivariant cobordism group of surface diffeomorphisms $\Omega_G$ to $B_G$. We will prove that the kernel of this surjection is isomorphic to $H_2(G;Z)$. Thus $\Omega_G$ is an Abelian group extension of $B_G$ by $H_2(G;Z)$. Finally we will prove that the group $B_G$ contains only elements of order two if and only if every complex character of $G$ has values in $R$. This property shows a strong relationship between the functor $B$ and the representation theory of finite groups.

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