#### Volume 8, issue 3 (2004)

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Morita classes in the homology of automorphism groups of free groups

### James Conant and Karen Vogtmann

Geometry & Topology 8 (2004) 1471–1499
 arXiv: math.QA/0406389
##### Abstract

Using Kontsevich’s identification of the homology of the Lie algebra ${\ell }_{\infty }$ with the cohomology of $Out\left({F}_{r}\right)$, Morita defined a sequence of $4k$–dimensional classes ${\mu }_{k}$ in the unstable rational homology of $Out\left({F}_{2k+2}\right)$. 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 $Out\left({F}_{4}\right)$. 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 $Aut\left({F}_{r}\right)$.

##### Keywords
automorphism groups of free groups, graph homology
##### Mathematical Subject Classification 2000
Primary: 20J06
Secondary: 20F65, 20F28