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2004 Morita classes in the homology of automorphism groups of free groups
James Conant, Karen Vogtmann
Geom. Topol. 8(3): 1471-1499 (2004). DOI: 10.2140/gt.2004.8.1471


Using Kontsevich’s identification of the homology of the Lie algebra 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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James Conant. Karen Vogtmann. "Morita classes in the homology of automorphism groups of free groups." Geom. Topol. 8 (3) 1471 - 1499, 2004.


Received: 21 June 2004; Revised: 1 December 2004; Accepted: 24 November 2004; Published: 2004
First available in Project Euclid: 21 December 2017

zbMATH: 1121.20036
MathSciNet: MR2119302
Digital Object Identifier: 10.2140/gt.2004.8.1471

Primary: 20J06
Secondary: 20F28, 20F65

Rights: Copyright © 2004 Mathematical Sciences Publishers


Vol.8 • No. 3 • 2004
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