Abstract
This chapter discusses characteristic classes of surface bundles and bounded cohomology. The isomorphism class of a surface bundle is completely determined by the induced homomorphism. If Σg is a closed orientable surface of genus g, then a differentiable fibre bundle π:E→X, whose fibre is diffeomorphic to Σg, is called a surface bundle or a Σg- bundle. The characteristic classes of surface bundles are highly nontrivial. The chapter presents a sufficient condition for the characteristic classes of surface bundles to vanish. An abstract group is called amenable, if there exists a left invariant mean on the set of all bounded real valued.
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