| Using Kontsevich’s identification of the homology of the Lie algebra l∞ with the
cohomology of Out(Fr ), Morita defined a sequence of 4k-dimensional classes µk
in the unstable rational homology of Out(F2k+2 ). He showed by a computer
calculation that the first of these is non-trivial, so coincides with the unique
non-trivial rational homology class for Out(F4 ). Using the “forested graph complex”
introduced in [2], we reinterpret and generalize Morita’s cycles, obtaining
unstable cycles for every odd-valent graph. The description of Morita’s original
cycles becomes quite simple in this interpretation, and we are able to show that
the second Morita cycle also gives a nontrivial homology class. Finally, we view
things from the point of view of a different chain complex, one which is associated
to Bestvina and Feighn’s bordification of outer space. We construct cycles
which appear to be the same as the Morita cycles constructed in the first part
of the paper. In this setting, a further generalization becomes apparent, giving
cycles for objects more general than odd-valent graphs. Some of these cycles lie
in the stable range. We also observe that these cycles lift to cycles for Aut(Fr ). |