  There is a series of cycles in the rational homology of the groups Out(F_n),
first discovered by S. Morita, which have an elementary description in terms of
finite graphs. The first two of these give nontrivial homology classes, and it
is conjectured that they are all nontrivial. These cycles have natural lifts to
the homology of Aut(F_n), which is stably trivial by a recent result of
Galatius. We show that in fact a single application of the stabilization map
from Aut(F_n) to Aut(F_(n+1)) kills the Morita classes, so that they disappear
immediately after they appear.
