Ralph Grieder | Searcharxiv

arXiv1999

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

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 $Ω_G$ to $B_G$. We will prove that the kernel of this surjection is isomorphic to $H_2(G;Z)$. Thus $Ω_G$ 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.

math.AT↗

arXiv1998

Geometric Representation Theory and G-Signature

Let G be a finite group. To every smooth G-action on a compact, connected and oriented 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. We will show that the map which sends G to B_G is functorial and carries many features of the representation theory of finite groups and thus describes a geometric representation theory. We will prove that B_G consists only of copies of Z and Z/2Z. Furthermore we will show that there is a surjection from the G-equivariant cobordism group of surface diffeomorphisms to B_G. We will define a G-signature which is related to the G-signature of Atiyah and Singer and prove that this new G-signature is injective on the copies of Z in B_G.

math.AT↗