Cohomological obstructions to Nielsen realization
J. Topol. 8 (2015) 2, 352–376.
For a manifold M and a subgroup G < Mod(M) of the mapping class group, the question of whether G can be lifted to the diffeomorphism group Diff(M) is an example of a Nielsen realization problem. This question is related to an existence question for flat connections on fiber bundles with with monodromy G. In this paper we consider the case M is a locally symmetric manifold with a basepoint and G is the point-pushing subgroup. The simplest instance – when M is a closed orientable surface of genus at least 2 – was worked out by Bestvina-Church-Souto, who showed a lift does not exist. We give new cohomological techniques to generalize their result to higher dimensional locally symmetric manifolds. The main tools include Chern-Weil theory, Milnor-Wood inequalities, and Margulis superrigidity.