Geometric Representation Theory and G-Signature

1998/11/17 by Ralph Grieder, Grieder, Ralph
Mathematics · #20C15 #57R85 #57S25 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.AT #math.GR #math.GT #msc:20C15 #msc:57R85 #msc:57S25

paper · pdf · doi:10.48550/arxiv.math/9811102

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

arxiv created 1998/11/17 · arxiv updated 2009/11/30

Abstract

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.

Related

• G-Actions on Riemann Surfaces and the associated Group of Singular Orbit Data
• Group actions on Jacobian varieties
• On Dihedral Group Actions on Riemann Surfaces
• Canonical representations of surface groups
• Topological components of surface group representations and signature