Geometry & Topology, Vol. 8 (2004) Paper no. 40, pages 1471--1499.

Morita classes in the homology of automorphism groups of free groups

James Conant, Karen Vogtmann

Abstract. Using Kontsevich's identification of the homology of the Lie algebra l_infty with the cohomology of Out(F_r), Morita defined a sequence of 4k-dimensional classes mu_k in the unstable rational homology of Out(F_{2k+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(F_4). Using the "forested graph complex" introduced in [Algebr. Geom. Topol. 3 (2003) 1167--1224], 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(F_r).

Keywords. Automorphism groups of free groups, graph homology

AMS subject classification. Primary: 20J06. Secondary: 20F65, 20F28.

DOI: 10.2140/gt.2004.8.1471

E-print: arXiv:math.QA/0406389

Submitted to GT on 21 June 2004. (Revised 1 December 2004.) Paper accepted 24 November 2004. Paper published 5 December 2004.

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James Conant, Karen Vogtmann
Department of Mathematics, University of Tennessee
Knoxville, TN, 37996, USA
Department of Mathematics, Cornell Univeristy
Ithaca, NY 14853-4201, USA

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