Some groups of mapping classes not realized by diffeomorphisms

  • Mladen Bestvina

    University of Utah, Salt Lake City, USA
  • Thomas Church

    Stanford University, USA
  • Juan Souto

    Université de Rennes 1, France


Let Σ\Sigma be a closed surface of genus g2g\ge 2 and zΣz\in\Sigma a marked point. We prove that the subgroup of the mapping class group Map(Σ,z)\mathrm{Map}(\Sigma,z) corresponding to the fundamental group π1(Σ,z)\pi_1(\Sigma,z) of the closed surface does not lift to the group of diffeomorphisms of Σ\Sigma fixing zz. As a corollary, we show that the Atiyah–Kodaira surface bundles admit no invariant flat connection, and obtain another proof of Morita’s non-lifting theorem.

Cite this article

Mladen Bestvina, Thomas Church, Juan Souto, Some groups of mapping classes not realized by diffeomorphisms. Comment. Math. Helv. 88 (2013), no. 1, pp. 205–220

DOI 10.4171/CMH/283