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Register a new userGeometric Representation Theory and G-Signature
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Authors
Ralph Grieder
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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.
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Johnsons characterization of amenable groups states that a discrete group $Gamma$ is amenable if and only if $H_b^{n geq 1}(Gamma; V) = 0$ for all dual normed $mathbb{R}[Gamma]$-modules $V$. In this paper, we extend the previous result to homomorphisms by proving the converse of the Mapping Theorem: a surjective homomorphism $phi colon Gamma to K$ has amenable kernel $H$ if and only if the induced inflation map $H^bullet_b(K; V^H) to H^bullet_b(Gamma; V)$ is an isometric isomorphism for every dual normed $mathbb{R}[Gamma]$-module $V$. In addition, we obtain an analogous characterization for the (smaller) class of surjective homomorphisms $phi colon Gamma to K$ with the property that the inflation maps in bounded cohomology are isometric isomorphisms for all normed $mathbb{R}[Gamma]$-modules. Finally, we also prove a characterization of the (larger) class of boundedly acyclic homomorphisms $phi colon Gamma to K$, for which the restriction maps in bounded cohomology $H^bullet_b(K; V) to H^bullet_b(Gamma; phi^{-1}V)$ are isomorphisms for suitable dual normed $mathbb{R}[K]$-module $V$. We then extend the first and third results to spaces and obtain characterizations of amenable maps and boundedly acyclic maps in terms of the vanishing of the bounded cohomology of their homotopy fibers with respect to appropriate choices of coefficients.
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