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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 an earlier paper, we reinterpret and generalize Morita's cycles,
obtaining an unstable cycle for every connected odd-valent graph. (Morita
has independently found similar generalizations of these cycles.)
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).
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