Title: (Un)boundedness of characteristic classes of manifold bundles
Speaker: Sam Nariman (Purdue University)
Abstract: The plane bundles over surfaces are classified by their Euler class and Milnor showed that if they admit flat structures, the Euler class should be a bounded class. This result has been generalized in many directions and in particular for a connected finite-dimensional Lie group G, it has been extensively studied the conditions under which the characteristic classes of flat principal G-bundles are bounded classes viewed in the cohomology of the classifying space of the group G with the discrete topology. In this talk, I will talk about a joint work with N.Monod, in which we started studying the same question for diffeomorphism groups in different regularities that are infinite-dimensional topological groups. In particular, we computed the bounded cohomology of Diff_0(S^1) and Diff_0(D^2) entirely and showed that they are isomorphic to the polynomial ring generated by the Euler class. We also answered Ghys’ question about generalizing Milnor-Wood inequality to flat S^3-bundles by showing that the Euler class is unbounded in H^4(Diff_0(S^3);R).
URL associated with Seminar: https://sites.google.com/view/topoandggt/home
