Cohomological obstructions to Nielsen realization

@article{Tshishiku2014CohomologicalOT,
  title={Cohomological obstructions to Nielsen realization},
  author={Bena Tshishiku},
  journal={Journal of Topology},
  year={2014},
  volume={8}
}
For a based manifold (M,*) , the question of whether the surjective homomorphism Diff(M,*)→π0Diff(M,*) admits a section is an example of a Nielsen realization problem. This question is related to a question about flat connections on M‐bundles and is meaningful for M of any dimension. In dimension 2, Bestvina–Church–Souto [‘Some groups of mapping classes not realized by diffeomorphisms’, Comment. Math. Helv. 88 (2013) 205–220] showed a section does not exist when M is closed and has genus g⩾2… 
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