It is well known that the first cohomology of the mapping class group M(S) of a connected closed orientable surface S of genus g is trivial. In this note, we prove that this result holds for any subgroup G in M(S) containing the Torelli group, provided g is at least three. We also provide counterexamples to this result in genus two. These results provide a partial answer to a question of Ivanov.
This paper is to appear in Topology.
Contact: mccarthy@math.msu.edu
Last Revised 7/9/99