On representability of the smooth Euler class

摘要

The Euler class, which lies in the second cohomology of the group of orientation preserving homeomorphisms of the circle, is pulled back to the "smooth" Euler class in the cohomology of the group of orientation preserving smooth diffeomorphisms of the circle. Suppose a surface group Gamma (of genus > 1) is a normal subgroup of a group G, so that we have an extension of Q = G/Gamma by Gamma. We prove that if the canonical outer action of Q on Gamma is finite, then there is a canonical second cohomology class of G restricting to the Euler class on Gamma which is smoothly representable, that is, it is pulled back from the smooth Euler class by a representation from G to the group of diffeomorphisms. Also, we prove that if the above outer action is infinite, then any second cohomology class of G restricting to the Euler class on Gamma is not smoothly representable.